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ELEMENTS (J?L 

^.0 83 

of 


NATURAL PHILOSOPHY. 


BY 


W. H. C. BARTLETT, LL. D., 

FiOFSASOR OF NATURAL. AND EXPERIMENTAL PHILOSOPHY IN THE UNITED STATES 
MILITARY ACADEMY AT WEST POINT. 


I I.-A COUSTICS. 
I I I.-O P T I C S. 



FIFTH EDITION, REVISED AND CORRECTED. 


NEW YORK: 

PUBLISHED BY A. S. BARNES & CO., 

t 

111 & 113 William Street (cor. John). 

1868. 






\ 


Entered according to Act of Congress, in tlie year One Thousand 
Eight Hundred and Fifty-two, 

By W. II. C. BARTLETT, 

in the Clerk’s Office of the District Court of the United States for the Southern District 

of New-York. 


T ransier 

Engineers School Liby. 
{ June 29,1931 



f . P. JONES <fc CO., STEREOTYPEB8, 
183 William-street. 


G. W. WOOD, PRINTER, 
81 John-st., cor. Dutch. 








PREFACE. 


Those who are familiar with the subjects of which the 
present volume professes to treat, will readily recognize the 
sources whence most of its materials are drawn. In the use 
of these materials, no distinction of principle is made between 
sound and light. Both are regarded and treated as the 
effects of certain disturbances of that particular state of mole¬ 
cular equilibrium which determines the ordinary condition of 
natural bodies; the only difference being in the n^edia through 
which these disturbances are propagated, and in the organs 
of sense by which their effects are conveyed to the mind. 
The study of Acoustics is, therefore, deemed to be not only a 
useful, but almost a necessary preliminary to that of Optics. 

In the preparation of the part relating to Sound, great use 
was made of the admirable monograph of Sir John IIerschel, 
published in the Encyclopedia Metropolitana; and whenever 
it could be done consistently with the plan of the work, no 
hesitation was felt in employing the very language of that 



4 


PREFACE. 


eminent philosopher. Much valuable matter was also drawn 
from Mr. Airy’s Tracts, and from the labors of Mr. Robison 
and M. Peschel. 

In addition to the works of the authors just cited, those of 
Mr. Coddington, Mr. Powell, Mr. Lloyd, Sir David Brew¬ 
ster and M. Babinet were freely consulted in constructing 
the part relating to Optics. 



I 


TABLE OF CONTENTS. 


ACOUSTICS. 

Introductory Remarks and Definitions, 

Waves in general, ..... 
Velocity of Sound in Aeriform Bodies, 

Velocity of Sound in Liquids, 

Velocity of Sound in Solids, .... 

Pitch, Intensity, and Quality of Sound, 

Siren, ....... 

Divergence and Decay of Sound, 

Molecular Displacement, ..... 

Interference of Sound, .... 

New Divergence and Inflexion of Sound, . 

Reflexion and.Refraction of Sound—Echos, 

Hearing Trumpet, ..... 

Whispering Galleries and Speaking Tubes, 

Musical Sounds, ...... 

Vibrations of Musical Strings, 

Monochord, ...... 

Vibrating Columns of Air, 

Vibrations of Elastic Bars, .... 

Vibrations of Elastic Plates and Bells, 

Communication of Vibrations, .... 

The Ear, ...... 

Music, Chords, Intervals, Harmony, Scale, and Temperament, 
Table of Intervals, ..... 

Table of Intervals with the Logarithms, 


Page 

9 

20 

24 

38 

44 

47 

49 

54 

58 

61 

70 

75 

90 

91 

94 

97 

Ill 

114 

126 

128 

131 

135 

137 

143 

154 








<3 


TABLE OF CONTENTS. 


OPTICS. 

Tage 

Introductory Remarks and Definitions, 

167 

Reflexion and Refraction of Light, 

. . 171 

Table of Refractive Indices and Refractive Powers, 

. 176 

Deviation of Light at Plane Surfaces, 

176 

Deviation of Light at Spherical Surfaces, 

188 

Deviation of Light by Spherical Lenses, 

201 

Deviation of Light by Spherical Reflectors, 

. 211 

Spherical Aberration, Caustics and Astigmatism, . 

216 

Oblique Pencil through the Optical Centre, 

220 

Optical Images, ..... 

222 

The Eye and Vision, ..... 

232 

Microscopes and Telescopes, 

. . 238 

Common Astronomical Telescope, 

. 245 

Galilean Telescope, .... 

246 

Field of View, ..... 

. 248 

Terestrial Telescope, .... 

250 

Compound Refracting Microscope, . . . 

252 

Astronomical Reflecting Telescope, 

253 

Gregorian Telescope, .... 

. . 254 

Cassegrainian Telescope, .... 

256 

Newtonian Telescope, .... 

. 25l 

Dynameter, ..... 

251 

Camera Lucida, ..... 

. . 261 

Camera Obscura, .... 

263 

Magic Lantern, ..... 

264 

Solar Microscope, .... 

265 

Chromatics, ...... 

. 266 

Color by Interference, Color of Gratings, . 

267 

Table of Wave Lengths, .... 

. 276 

Colored Fringes of Shadows and Apertures, 

282 

Colors of Thin Plates, .... 

. 284 

Colors of Inclined Glass Plates, 

290 

Colors of Thick Plates, . . . 

. 291 










TABLE OF CONTENTS. 
- t 

Color from Unequal Refrangibility, 

Dispersion of Light, ..... * 

Table of Dispersive Powers, . . 

Chromatic Aberration, ..... 

Achromatism, . . . . . 

Internal Reflexion, . . 

Absorption of Light, ..... 

The Rainbow, ...... 

Halos, ....... 

Polarization of Light, ..... 

Polarization by Reflexion and Refraction, . 

Polarization by Absorption, ..... 

Double Refraction, ..... 

Circular Polarization, . . . ... . 

Chromatics of Polarized Light, . . . 

.... ^ 

** . A.*w**- 


7 

Page 

293 

298 

301 

302 

305 

309 

311 

314 

321 

322 

327 

334 

335 

345 

348 



























































* • 


* 

A • 










w 
















) 




- i 




















































































































































ELEMENTS OF ACOUSTICS. 


§1. The principle which, connects ns with the external 
world through the sense of hearing, is called soum>; and sound, 
that branch of Natural Philosophy which treats of sound, 

. , . Acoustics. 

is called Acoustics. 

To explain the nature of sound, the laws of its propaga¬ 
tion through the various media which convey it to our 
ears, the mode of its action upon these organs, the modifi- objects of acous. 
cations of which sound is susceptible in speech, in music tics - 
and in unmeaning noise, as well as the means of pro¬ 
ducing and regulating these modifications, are the objects 
of acoustics. 

§ 2. All impressions derived through the senses, imme- 0UT 

diately follow and may, therefore, be said to arise from 
peculiar conditions of relative motion among the elements 
of which certain parts of our physical organization are 
constructed. These conditions are mainly determined by Condition810 

. cause sensation 

the internal state of the bodies with which we are in sen- determined, 
sible contact; and it is entirely from the transfer of work, 
in the form of molecular living force , from them to our 
organs of sense, that all impressions from the external 
world arise. This transfer is unaccompanied by transfer 
of material, and the agents are the molecular forces that 
determine the physical condition, and, therefore, the sensi¬ 
ble qualities of all bodies. 

§ 3. We have already referred, in the introduction to 
the first volume, to Boscovich’s views upon this subject, 
and shall now give some illustration of the mode in which, 



10 


NATURAL PHILOSOPHY. 


Exponential 

curve; 


Attractive 
ordinates; 
Repulsive 
ordinates; 


Neutral points; 

Temporary 
molecule; 

Permanent 
molecule; 

When 

permanence 

exists. 


according to that distinguished philosopher, all bodies are 
formed. 

For this purpose let us resume the exponential curve as 
exhibited in the annexed figure, and which Boscovich sup¬ 



poses to represent the law and intensity of the action of 
one atom of a body upon another. We have seen that the 
ordinates of those portions of the curve which lie above 
the line A O, denote the attractive, while the ordinates of 
the portions below, represent the repulsive energies of an 
atom A for another atom situated anywhere upon this 
line. That at the points C\ D', C'\ in which the curve 

intersects the line A C, the reciprocal action of, the atoms 
reduces to nothing, and the atoms become neutral. Also 
that an atom situated at D\ D" or D” r and the atom A 
constitute a temporary molecule , while the molecule formed 
of the atoms A and C\ A and G'\ or A and C"\ has a cer¬ 
tain degree of permanence, resisting compression and dila¬ 
tation, and tending to regain its original bulk when the 
distending or compressing cause is withdrawn. But this 
permanence only obtains when the disturbing force is such 
as to change the interval between the atoms by a distance 
less than that which separates the consecutive positions of 
neutrality; for if the molecule A G\ for example, be com¬ 
pressed into a less room than A D\ the atom originally at 
G'\ will not return to that point, but will be attracted by 
A, and the molecule will tend to collapse into the bulk 
A O'. If A C" be stretched beyond the bulk A D”, it 
will tend to take the dimension A O'". The onl^mole- 
cule that cannot be permanently changed by compression 
is A O'. 







ELEMENTS OF ACOUSTICS. 


11 


The component atoms of molecules thus constituted are, 
when in a state of relative equilibrium, in a condition of 
inactivity upon each other. The approximation or sepa- How the recipro 
ration of the atoms by the application of some extraneous ^^ i ^^“ ons 
cause, gives rise to the exertion of the repulsive or at- cited, 
tractive forces inherent in the atoms, and thus these forces 
may be said to be excited or brought into action. The 
compression or dilatation is the occasion , not the efficient 
cause of the attractions and repulsions among the atoms. 



Fig. 3. 



§ 4. The intensity of the atomical forces determines the Form of the es- 
form of the exponential curve. If a ^ g determined, 

very moderate force produce a sensi¬ 
ble displacement of the atoms, the 
ordinates E' d\ and Ed , on each side 
of the position C\ of inactivity, must 
be short, and the exponential curve will cross the axis very 
obliquely, in order that the ordi¬ 
nates expressing the attractive and 
repulsive forces may increase slowly. 

If, however, it require great force 
to produce a sensible compression or 
dilatation, the curve must cross the 
axis almost perpendicularly. But in 

every case it must be remarked, and the remark is most small compres- 
important, that when the compression or distension bears and di8teD ‘ 
a small proportion to the distance between the neutral 
positions of the atoms, the degree of compression or dis¬ 
tension will be sensibly propor¬ 
tional to the intensity of the dis¬ 
turbing force. For, when the 
displacement D’ E or D' E' is 
very small in comparison to 
G' D\ the elementary arc dD r d' 
will sensibly coincide with a 
straight line, and the ordinates 

E d and E' d\ be proportional to the compression D r E 
or distension D’ E’. That is to say, because action and 


Fig. 4 




$1V’ F' 











12 


NATURAL PHILOSOPHY. 


Their reaction are equal, a disturbed 

consequences. wiR fa urged fa^ to _ 

wards its position of neutrality 
by a force whose intensity is 
proportional to the distance of 
the atom from that point. 

Moreover, supposing the atom 
-4, Fig. 4, to be kept station¬ 
ary, and the points E^ and E\ to mark the limits of 
the disturbance of the other atom, this latter will return 
to its position of neutrality D\ with a living force due to 
the action of the force of restitution over the path E D\ 
or E' JD r ; it will, therefore, pass the point D\ after which 
the direction of the action will be reversed, the living 
force will be destroyed, the atom will again return to its 
Perpetual osdiia- position of neutrality, which it will pass as before, and for 
a™; the same reason, and thus be kept in perpetual oscillation. 

But the action between the two atoms of the molecule be¬ 
ing reciprocal, the atom A will not remain stationary, but 
will move in the same direction as the disturbed atom and 
tend to preserve its neutral distance, and the oscillation 
checked. that would otherwise continue will, therefore, be checked. 

Action of the sim- § 5. Let us next take the case of a molecule of the sim¬ 
plest molecule on pl e st constitution, to wit, one composed of two atoms, and 

an atom. . . 7 . _ 7 

examine its action on a third atom situated on the prolon¬ 
gation of X y, joining its elements. 



rig. 5. 



First case; 


Suppose a molecule X X, composed of the two atoms 
X and ]T, which are placed, the former at M, and the lat- 
















13 



ELEMENTS OF ACOUSTICS. 


ter at the last limit of cohesion C\ Fig. 1. The dotted and Ex P° nential 
waving curve beginning at Y and running towards (7, will component 
represent the exponential curve of the atom X, in that atoms; 
direction, while the similar curve beginning at the point 
X, will represent that of the atom X, and the full curve 
^C' A' D’ R’C" A" &c., of which the ordinate corres¬ 
ponding to any point of the line A (7, is equal to the alge¬ 
braic sum of the ordinates of the dotted curves correspond¬ 
ing to the same point, will be the exponential curve of the That of the 
molecule X Y\ and will give the action of the molecule molecu,e; 
upon a third atom placed any where on the line A C be¬ 
yond Y. The curve has been carefully constructed ac¬ 
cording to the conditions of the case, and shows by simple 
inspection how different the action of even the simplest 
molecule is from that of a single atom. The neutral posi- Neutral position* 
tions of an atom with respect to this molecule will be at ^p" c “ t ®“ b '! s lth 
A , G\ D” and so on to G. A curve having a molecule; 

cusp at A, the middle point of the distance X Y , and 
diverging so as to be asymptotic with the lines c b and c r b\ 
will give the law and intensity of the action on an atom 
situated between X and Y. 

§ 6. If instead of placing the atoms at a distance apart second case, 
equal to that of the last limit of cohesion from A, as in the 
last case, we had supposed them separated by the distance 
A C ", Fig. 1, the resulting exponential curve would have 
been still more unlike that of a single atom; for in that case 



several of the attractive branches, Fig. 6, of one of the atomi¬ 
cal curves would have stood opposed to the repulsive ResuIting act " ,a 

x x x # on an atom. 

branches of the other, and the molecule thus rendered in- 







34 


NATURAL PHILOSOPHY. 


Esempllfi cation. 


Third case; 


Construction; 


Construction of 
the exponential 
curve giving the 
action of a 
molecule on an 
atom. 


active on a third atom till the 
latter be removed nearly to the 
furthest limit of the scale of 
corpuscular action. This third 
atom will, therefore, admit of 
considerable latitude of displace¬ 
ment without much opposition 
or any great effort to regain its 
primitive position; a fact we 
often see exemplified in the class 
of liquid bodies. 

§ 7. Let us now take the 
molecule composed of two atoms 
placed at the limits A and C ", 

Fig. 1, and examine its action on 
a third atom somewhere on 
the line B B\ which bisects at 
right angles the distance A C". 
Suppose the third atom placed 
at z. Join z with A and C'\ and 
construct the single atomical 
curves of A and C" in reference 
to s, and suppose the atom z in 
Fig. 7, to have a position with re¬ 
spect to A and C’\ correspond¬ 
ing to any position between B” 
and C"\ Fig. 1; thus situated, it 
will be repelled both by A and 
<7", Fig. 7. In a pair of dividers 
take the ordinate z m, Fig. 1, and 
lay it off from s, on the prolong¬ 
ations of Az and C" z, Fig. 7, 
and construct the parallelo¬ 
gram z m n m’\ the diagonal 
z n, will represent in direction 
and intensity the action of the 
molecule A C" on the third atom. 


Fig. 7. 



Draw a perpendic- 








ELEMENTS OF ACOUSTICS. 


15 


ular to B B' through the point 2 , and take the distance 
z R" equal to z n , the point R" will be one point of the 
exponential curve of the molecule AC" in the direction 
B B r . Other points being determined in the same way, the 
waved lines of Fig. Y will indicate the action sought; 
the ordinates of the branches A\ A!\ A"\ &c., on one side 
of B B\ denoting attractions, while those of the branches 
R\ R'\ R"\ &c., on the opposite side, denote repulsions. 

We see that this action differs remarkably from that of Action differs 
a single atom. The curve has, to be sure, like that of a thal 

single atom, many alternations of attractions and repul¬ 
sions, but these alternations become less marked as they 
approach the molecule; and instead of insuperable repul¬ 
sion at the greatest vicinity 7", we find there a neutral 
point. Moreover, in approaching the molecule, the repul¬ 
sive action ceases at D\ where attraction begins and con¬ 
tinues, so far as there is any action, all the way through 
to JD r on the opposite side of A C". This molecule is ever 
active when approached along the line B B\ except at 
certain neutral positions where the direction of the action 
is reversed, and is easily penetrable in this direction, 
whereas along the line A C" it exerts little or no action 
within certain limits, and is capable of an infinite repul¬ 
sion within its last limit of cohesion. Thus we see that 
even in this simplest constitution of a molecule, the action 
on an atom is susceptible of great variety by mere diffe¬ 
rence of position and distance between its component 
atoms; and it would be easy to show that while the law 
of the atomic action in all bodies is the same, the reci- ^ ofatomic 
procal action of the molecules com¬ 
pounded of these atoms may be un¬ 
speakably various according to the 
relative position and distance of the 
component atoms. 

§ 8. Confining, for the present, 
the motion of the third atom to the 
plane of the lines A C" and B B\ 


Fig. a 



action the same 
in all bodies. 
Reciprocal action 
of molecules 
infinitely various. 





16 


NATURAL PHILOSOPHY. 


Action of a 
molecule on an 
atom. 


Constitution of 
an elementary 
surface. 


Oscillation of th< 
disturbed atom; 


That of the 
atoms of the 
molecule. 


Fig. 7, we see that when it is at 2 , FJg> a 

it is repelled by the molecule AC"; 
when at z' it is attracted, and 
the action is reduced to nothing 
at the point D". "When the atom is 
drawn aside from its neutral position 
_Z>", say to 2 ", Fig. 8, it will he re¬ 
pelled by C" and attracted by A, 
because the distance from the former 
will he diminished, while that from A will he increased. 
Take z" h to represent the intensity of the repulsion and 
z" 0 that of the attraction; complete the parallelogram 
0 2 " 7i q , and we shall find the molecule urged to its neu¬ 
tral position D" by a force whose intensity and direction 
are represented by the diagonal z" q; so that, so far as the 
action in the plane AC" D" is concerned, D" is a posi¬ 
tion of stable equilibrium, and the three atoms A, C" and 
D" will constitute for moderate displacements a permanent 
molecule, presenting an elementary surface haying length 
and breadth. The same would be true were the third 
atom placed at D r or D"\ &c., Fig. 7. 

1 The disturbed atom when at z" being urged back to its 
place of neutrality by the molecule A C'\ will reach that 
point with a certain amount of living force, due to the ac¬ 
tion of the force of restitution over the path from z” to D"; 
it will, therefore, pass to the opposite side of D", where 
the action being in the opposite direction, its living force 
will be destroyed, after which it will be brought back and 
made to oscillate about D" as long as A and C" are star 
tionary. But while the third atom is on the side 2 ", that 
at C" will be repelled by it, and that at A attracted; the 
contrary will be the case when the atom is on the oppo¬ 
site side from z", so that the atoms of the molecule A C n 
will also oscillate, and obviously in such manner as to 
to cause the neutral position to follow the displaced 
atom. 



Explanation of 
figure; 


9. Now conceive a triangle A C" C each of whose 





ELEMENTS OF ACOUSTICS. 


17 


sides is equal to a distance at which 
two atoms may form a permanent 
molecule, and suppose an atom to be 
placed at each vertex; these atoms 
will form a permanent molecule. 

Place a fourth atom at the vertex 
D'\ of a pyramid of which the base 
is the elementary plane formed by 
the first three atoms, and each of the 
edges about the vertex is equal to a 
distance necessary for two atoms to 
form a permanent molecule. It will be obvious, from 
what has already been said, that the fourth atom or that 
at the vertex cannot be disturbed without being resisted Permanent 
and urged back to its neutral place by the action of the ™^ uleoffour 
molecules which form the base; for, if it be moved aside 
in either of the plane faces of the pyramid, it will, § 8, 
be opposed by the force of restitution due to the action of 
the molecule of two atoms in the same plane; and if 
moved out of these planes, its distance from one at least 
of the atoms in the triangular base must be altered, thus 
exciting a force of restitution. What has been said of the 
atom at the vertex of the pyramid is equally applicable 
to each of those in the base when considered in reference 
to the three others, and hence the four atoms A y C", 

6'/', D'\ form a permanent molecule ; and from its capa¬ 
bility to resist the approach of a fifth atom, another mole¬ 
cule or particle, in every direction, we derive the idea of 
an elementary solid, having length, breadth and thickness. Elementary solid. 
A disturbance of any one of the four atoms will put the Disturbance w5U 

*1 . " * ... cause the neutral 

others in motion, and it will appear on the slightest con- points to follow 
sideration that the directions of these motions will be such thedistnrbcd 

atoms. 

as to cause the neutral positions to shift in the direction of 
the atoms which have been disturbed from them. 

§10. What has been said of the action of atoms to Same reasoning 
form molecules may easily be shown to be true of the 3ecuies and 
reciprocal action of molecules to form particles, and ofP articlea - 
2 






18 


NATURAL PHILOSOPHY. 


Red rocal action 
of elements 
confined to small 
distances. 


The most subtile 
body 

conceivable; 


When it would 
take its 

permanent form: 
fitier. 


Structure of the 
atmosphere, 


particles to form tlie bodies which, in all their endless 
variety of physical characters, come within the reach of 
onr senses. And according to this view, the characteris¬ 
tic peculiarities of all bodies are to be understood as aris¬ 
ing solely from differences in the action which their atoms, 
molecules and particles are capable of exerting on each 
other, and upon those of the bodies with which they may 
be brought into sensible contact. 

But it must be remarked that all these differences of 
action are confined to small and insensible distances which 
lie within the limits of physical contact. At all consider¬ 
able distances we find nothing but the action of gravita¬ 
tion, of which the intensity is proportional to the number 
of atoms or the mass directly, and to the square of the 
distance inversely. 

§ 11. The most subtile and attenuated body of which we 
can form any conception, according to Boscovich, is one 
composed of atoms arranged at distances from each other 
equal to that which determines the furthest limit of cohe¬ 
sion, or that beyond which gravitation begins. But such 
a body, when abandoned to itself, would shrink into 
smaller dimensions in consequence of the gravitating force 
between the atoms not adjacent to each other, and the 
contraction would continue till the repulsions which it 
would develope between the contiguous atoms had in¬ 
creased to an equilibrium with the compressing action, 
when the body would take its permanent form. 

Such we may suppose to be the constitution of that ethe¬ 
real medium which pervades all space, permeates every 
body, and connects us with the objects of whose existence 
we are made conscious through the sense of sight. 

§ 12. A body similarly constituted, but in which the 
atoms are replaced by molecules or particles arranged at 
distances less than the furthest limit of cohesion may give 
us an idea of the physical structure of our atmosphere. 
Here as in the last case the molecules cannot occupy their 



ELEMENTS OF ACOUSTICS. 


19 


neutral positions because of the forces of gravitation exist- its moiecuios 
ing between those molecules more remote from each other cannot be in 

° ,, . , tbeir neutral 

than the furthest limit of cohesion, which force will cause positions; 
the elements to crowd together; but we have seen that 
when the elements of a body are brought closer than those 
neutral positions which constitute permanence, the adjacent 
elements will repel, and can come to rest only when these 
antagonistic forces of attraction and repulsion balance. 

Add to these considerations the attraction of the earth 
for this fluid, and the equilibrium of any molecule will be conditions of it* 
found to result from the mutual balancing of the weight equilibrium, 
of the superincumbent column of molecules extending to 
the top of the atmosphere, and the repulsive action of 
the molecule in question for that immediately above it; 
and since the weight of the pressing column decreases as 
we ascend, the density must diminish in the same direc¬ 
tion, all of which we know to be confirmed by the indica¬ 
tions of the barometer. 

§ 13. Passing to the denser bodies, whether of the 
organic or inorganic class, as vegetable or animal tis¬ 
sue, water, clay, glass, gold, we find variety of structure 
without difference in the principles of aggregation. All ah bodies 
are built up of the same ultimate atomic elements, grouped 
into molecules, the molecules into particles, and the parti- atomic elements 
cles into the various bodies whose places in the scale of 
gradation, from the hardest to the softest solid, from the 
most viscous liquid to the most subtile gas, are deter- Their 
mined solely by the intensity, range and direction of^"^^ 6 
the atomic actions which mark their internal structure, determined 

§ 14. All bodies in nature are physically connected ah bodies 
with each other. Those plunged into the ocean are united 
by sensible contact with its common element. So of the 
bodies which exist in the atmosphere. The atmosphere Physics 
rests upon the ocean, and that ethereal medium which ZtZZa by 
permeates the atmosphere and the ocean, and extends 



20 


NATURAL PHILOSOPHY. 


Atmosphere, throughout all space, carries this connection to the hea- 
ether. venly bodies. 

Disturbance of a The disturbance of an atom, molecule or particle, will 

remitted alter its relative distances from the neighboring elements ; 
throughout the molecular forces on the side of the shortened distances 

6pac& will increase, while those on the opposite side will di¬ 

minish. The equilibrium which before existed will be 
destroyed, and the adjacent elements must also be dis¬ 
turbed ; these will disturb others in turn, and thus the 
agitation of a single element will be transmitted through¬ 
out space, and impart motion, in a greater or less degree, 
to the elements of all bodies. 


Motion affects 
the mind 
through organs of 
sense; 


All our 

impressions due 
to a common 
principle— 
motion. 


§ 15. Among the bodies thus affected are certain deli¬ 
cate and net-like ramifications of nervous tissue, which 
are spread over portions of our organs of sense. These 
nerves partake of the agitations transmitted to them from 
without, and by some mysterious process, call up in the 
mind impressions due to the external commotion. The 
structure and arrangement of these nerves differ greatly 
in the different organs, and while they are all subjected 
to the general laws which control the corpuscular action 
of bodies, yet each individual class is distinguished by 
peculiarities which determine them to appeal to the mind 
only when addressed in a particular way. We hear, feel 
and see by the operation of a common principle—motion; 
of this, there is endless variety in perpetual existence 
among the elements of the media in which we are im¬ 
mersed ; and, according as one or another of the organs 
of sense becomes involved in the particular motion adapted 
to excite the mind to action, will our sensation become 
that of sound, light, heat, or electricity. 


OF WAVES. 


All sensations 
dependent upon 
motion. 


§ 16. All sensations derived from our contact with the 
physical world depend, according to this view, upon the 



ELEMENTS OF ACOUSTICS. 


21 


state of relative motions among the elements of bodies; 

and we now proceed to consider those motions which are Those proper to 

suited to produce the sensation of sound, and we must be produce 800114 

careful to distinguish between the properties of solids and 

fluids in this respect. 

Conceive a perfectly homogeneous solid, that is, one in 
which the particles occupy the vertices of regular and 
equal tetrahedrons, and suppose its elements in a state 
of relative repose. A single particle being disturbed from 
its place of rest, through a very small distance, compared 
with the tetrahedral edges, will be urged back by the 
action of the surrounding elements with an energy which 
is, § 4, proportionate to the disturbance. This particle 0rbitofa 
will, when, abandoned to itself under these circumstances, part icic ; 
describe about its position of rest as a centre, an ellipse, 
or perchance, a circle or right line, the extreme varie¬ 
ties of the ellipse whose eccentricities are respectively 
zero and unity. Moreover, the time of describing a Time of 
complete revolution will, Mechanics, § 207, be constant, 
no matter what the size of the orbit within the limits sup¬ 
posed; and the mean velocity of the particle will, there- Mean velocity, 
fore, be directly proportional to the length of the orbit, 
or to any linear element of the same, as that of the semi- 
transverse axis. The disturbed particle being acted upon 
by its neighbours, these latter will experience from it the 
action of an equal and contrary force ; they must, there- Neighboring 
tore, move and describe similar orbits; and the same 8imi]arorblt8i 
will be true of the particles next in order, till the disturb- Disturbance 
ance becomes transmitted indefinitely. The disturbance transmittod ia 

d all directions. 

must take place in all directions from the primitive source, 
because the displacement of a single particle from its po¬ 
sition of rest breaks up the equilibrium on all sides ; and 
the disturbance must be progressive, since it is to an Disturbauce to 
actual displacement of a particle that the forces are due 
which give rise to the displacement in others.. It follows, 
therefore, that while the first disturbed particle is describ¬ 
ing its elliptical orbit the disturbance itself is being propa¬ 
gated from it in all directions, and that at the instant this 



22 


NATURAL PHILOSOPHY. 


First particle particle has completed one entire revolution, and begins 
having made one a second, the disturbance will have just reached another 
particle A 2 , in the distance, which particle will then be- 
nmve; gin for the first time to move, so that these two particles 

will during subsequent revolutions about their respective 
centres always be at the same angular distance from their 
starting points ; when the first particle A x has completed 
its second revolution, and the particle A 2 its first, the dis¬ 
turbance will have reached a third particle A 3 , still fur¬ 
ther in the distance, which begins its first revolution when 
a third begins to A 2 begins its second, and A , its third, and so on indefi- 
move - nitely. 

Space including Now, after the disturbance has reached the particle A 3 
particles in an ft j s plain that the particles 

positions in tlielr r _ A . 

orbits; between A l and A 2 mclu- Fig. 10 . 

sive will be in all possible 
situations in their respective 
orbits. For example, taking 

the instant in which A , first returns to its starting point, 
it will have described three hundred' and sixty degrees, 
the consecutive particle an arc less than this, the next par¬ 
ticle, in order, an arc still less, and so on till we reach A 2 , 
which will only just have begun to move. If then, we 
conceive a series of 
concentric spheres 
whose radii are re¬ 
spectively A 1 A 2 , 

AjA 3 , Al , A . 4 , &c. 
it is obvious that 
within the space in- 




Fig. 11. 


mustration. 


\ 


\ 


At _A% 

/ 

| *Aa 

1 * 



Explanation of c l u ded between these spherical surfaces, the particles will 

wave length. . x 1 x 

be m every possible stage of their circuits around their 
respective centres, and will, as we pass from surface to 
surface, be found moving in all possible directions in the 
planes of their several orbits; and the same would obvi¬ 
ously be true, if the radii of any two consecutive surfaces 
had been increased or diminished by the same length, the 
only difference being that the particles at the new position 







ELEMENTS OF ACOUSTICS. 


23 


of the surfaces, instead of being at the origin or places of 
rest from which they began their respective circuits, would 
occupy places more or less remote but equally advanced 
from these points. Thus, for example, had the radii been 
taken iji 3 + ii 2 i 3 , and A x A z + \ A 2 A Z , then wave length n<* 
would the particles at the new surfaces have been at an^JJ^J 10 aly 
angular distance from their respective places of primitive position, 
departure equal to 90°, but the surfaces would still have 
included between them in the direction of the radii, par¬ 
ticles in every possible state of progress in their circuits, 
the particle at the origin of departure being in this case 
at a distance from the surface of the smaller of the second 
set of spheres equal to three-fourths of the difference be¬ 
tween the radii of any two consecutive spheres of the first set. 

This particular arrangement of the particles of any 
body arising from the disturbance of one of its elements, 
and by which, after a certain lapse of time, all possible 
positions around their respective places of rest are occu¬ 
pied by the particles, in the order of succession, at the 
same time, is called a Wave. The distance, in the direction Wam 
of the radii, between any two of the consecutive spherical 
surfaces above described, is called the length of the wave. 

The term phase is used to express both the par- Pfcas*. 
ticular displacement and direction of the motion of a par¬ 
ticle in any wave. A wave lengthy therefore, is that interval wave length. 
of space which comprises particles in every possible phase. 

Particles which have equal displacements and motions, 
in the same direction, are said to be in similar phases; similar phases 
when the displacements and motions are equal and op¬ 
posite, the particles are said to be in opposite phases. 0 pposite phases 

The surface which contains those particles of a 
wave which are in similar phases, is called a wave front, wavefront; 

• In sound this last term will be used to denote the surface 
containing those particles which are, for the first time, in sound, 
beginning to move from their places of rest. 

In fluids the particles are not, as in solids, invariably 
connected, but admit of free motion among each other. 

When, therefore, a fluid particle is disturbed, it acts on 



24 


NATURAL PHILOSOPHY. 


Pulse. 


the surrounding particles as on detached masses, and 
having given up its motion after the manner of one body 
colliding against another, it comes to rest and continues 
so till disturbed again by some extraneous cause ; in the 
meantime, the surrounding particles move to assume with 
respect to it their positions of relative rest; other particles, 
more remote, partake in turn of this momentary move 
ment; one particle after another comes to rest, and thus, 
but a single wave, denominated a pulst, is transmitted 
throughout the medium. If, however, instead of aban¬ 
doning the fluid particle after impressing upon it its primi¬ 
tive motion, it were moved to and fro, like air before a 
vibrating spring, waves would succeed each other in fluids 
wave recurrence as in solids, the circumstances of wave recurrence being 
dependent on determined wholly by the action of the disturbing cause. 

A wave transmitted through any medium tends to 
throw the elements of all bodies which it meets in its 
course into a similar condition of wave motion. When 
the elements composing the nervous membranes of the 
ear become involved in certain of these motions, trans¬ 
mitted through the atmosphere or other medium with 
which the ear is in contact, we experience the sensation 
of sound ; when the nerves of the eye partake of a similar 
class of waving motions conveyed through the ether, we 
have the sensation of light; and when the waves are of 
that particular character to agitate the surface or cuta¬ 
neous nerves, the sensation becomes that of heat. 


Whence we 
experience the 
sensation of 
sound; 

Of light; 


Of heat. 


THE VELOCITY OF SOUND IN AERIFORM BODIES 


§ 17. Now, it is important to distinguish between the 
rate according to which the disturbance is propagated, 
velocity of wave and that with which each particle describes its orbit about 
vdTc^oTa lts pl ace °f rest - The first is called the wave velocity , the 

particle. second the velocity of the wave element. The first deter- 

deterlTles an mines the interval of time from the instant of primitive 
interval of time; disturbance to that which marks the beginning of motion 



ELEMENTS OF ACOUSTICS. 


25 


of any remote particle; the second, the quantity of action The second a 
communicated to this particle. The wave is but a form qn * ntlty of 
or shape , occurring, in the regular lapse of time, at places Wave is a form 
more and more remote from the place of first agitation, or sha P e - 
as from a centre, while the particles whose relative posi- Excursions of 
tions determine this form, never depart from their places partldesvcry 
of relative rest but by distances which are quite insigni-compared with 
ficant in comparison with the lengths of the waves. The a wave Ien ^ h - 
wave velocity is called the velocity of sound , of light, of 
heat, or of electricity , according to the sense to which the sound, of light, 
waves address themselves. We now proceed to investi¬ 
gate the velocity of sound, and shall begin with the 
aeriform bodies, taking the atmosphere first. 

From the definition of a wave, § 16, it follows that 
during the time in which the wave 


element, or single particle a, of air, Fig. 12 . 

describes one entire revolution in its 
orbit, the front of the wave will have 
progressed over the distance a a\ 
equal to a wave length. Denoting therefore, the wave 
velocity by V, the length of the wave a a', by A, and the 
time required for an element to make one complete cir¬ 
cuit by t , we shall have, Mechanics Eq. (6), 



(!)• 


Value for wave 
velocity. 


§18. The time t, is, as we have seen in Mechanics, The time t, 

§ 207, independent of the distance of the particle from its thTchamct^o? 
place of rest, and is determined by the acceleration due the disturbing 
to the intensity of the central force at the distance unity. foice * 

This intensity, in the case of sound, is the resultant of the 
antagonistic action of the force of disturbance and that of 
restitution, and as the latter is always constant for the 
same medium at the distance unity, or any other given 
degree of displacement, the value of t must result from 
the character of the disturbing force. Thus when the par- 





26 


NATURAL PHILOSOPHY. 


tide «, is made by any extraneous 
force to describe a path about its po- Fig. la 

sition of rest, the adjacent particles e . .j, 

niastration. d, e, will be thrown into motion, 

and will only return to their places ^ 

of departure after a. has been re¬ 
stored by the force of disturbance to d' 'e 

its position of rest; and since the 
places occupied at any instant by the particles b, <?, d , e , 
depend upon that of the particle a , the rate of motion of 
the former particles in their respective orbits, and there¬ 
fore the value of t , will be determined by the greater or 
less rapidity with which <z, is made to move under the 
action of the disturbing force. The motions of the parti¬ 
cles 5, <?, d , e , regulate in turn those of the next particles 
remote from the in order, and so on indefinitely, so that the disturbing 
force regulates the value of £, for all particles however 
remote from the primitive agitation at a. 


True for all 
particles however 


v independent g 19 . With the value of V it is not so: this is indepen- 

of the disturbing 

force. dent of the disturbing force. We have seen, § 12, that 

when in a state of relative rest, the elements of any me¬ 
dium are maintained in that condition by the opposing 
forces of repulsion between adjacent elements, and of 
attraction between those which are separated by a dis¬ 
tance greater than that which determines the furthest 
limits of corpuscular action. These forces are equal and 
opposite. Denote the sum of the re¬ 
pulsions of the particles which occupy Fi & m* 

a unit of surface by E. Conceive a . jl 

plane A B , passed through the me¬ 
dium, and the particles on the side 
X to be removed ; those distributed 
over a unit of surface of the opposite 
side will be pressed against the plane 
by a force equal to E\ and to keep 
the plane from moving would require 
the application of an equal and contrary force. But this 


Hiustration. 


X • 




ELEMENTS OF ACOUSTICS. 


27 


force, in the case of the atmosphere, is measured by the 
weight of a column of mercury whose base is unity, den¬ 
sity D ti , and height A, or by D n . g . A ; whence 


E = D n . A . g . (2). Measure of the 

elastic force of 
the atmosphere. 

The second member measures the Elastic force of the 
medium. 


A A' 


Fig. 15. 

V U 


§ 20. Let A B, G D, E Fj &c., be the positions of 
several strata of particles of air at 
rest and of which the molecular forces 
are in equilibrio ; and suppose them 
surrounded by a tube whose axis 
is perpendicular to their surfaces. 

If the stratum A B be moved by 
any extraneous cause towards the 
stratum C D, the latter will move under the action 
of the increased repulsion between it and the stra¬ 
tum A B. Suppose the stratum A B to take the position 
A! B\ at the instant the stratum C B begins to move. 
The distance A A\ will, from the views already given of 
the constitution of a fluid, be indefinitely small. 

Denote the distance A C by x ; A! G by x t ; and the 
elastic force exerted by the air in its state of rest on a unit 
of surface by Ej then supposing the cross section of the 
tube uniform, and its area equal to a, according to Mar- 
iotte’s law 

ax. : ax:: aE: aE\ 


Demonstration. 


Mariotte’s la^r. 


in which E t denotes the elastic force exerted by the air 
on a unit of surface between A’ B’ and G D ; whence 



Elastic force of 
the compressed 
air. 


The stratum G D is urged forward by the elastic force 










28 


NATURAL PHILOSOPHY. 


Moving force 
acting on a 
stratum; 


Ex and is opposed by the elastic force 
E; its motion will therefore be due to 


a(E—E) = aE ~—aE= aE‘ —— L 


Fig. 15. 

A A' C JB _ 


which is the moving force. And denoting the mass of 
the stratum CD by m, the acceleration due to this force, 
or the velocity generated in a unit of time, will be 


Velocity due to a x — X J 

this force; -- • —- j 

m x J 

and the velocity <y, generated in an elementary portion 
of time £, equal to that during which the stratum A B 
moves to the position A! B\ will be given by the relation 


a E x — x, . 

v =----. t. 

m X, 

Mechanics §55, which is obviously the velocity with 
which the stratum C D will be thrust from its state of 
velocity rest, and is analogous to that imparted to a stratum of fluid 
imparted to the pressed through an orifice in the bottom of a vessel con- 

Btratum CD. . . ° . 

taming a heavy fluid. 

The mass of the stratum CD will be the same whether 
we regard it concentrated into the plane CD , or ex¬ 
panded in both directions half way to the adjacent strata; 
in the latter case its volume would be a . x, and its den¬ 
sity a mean of the actual density of the whole fluid 
mass. The same being supposed of all the strata, the 
matter would become continuous; and denoting the 
mean density by D, we have 


Velocity in 
emf.ll time t; 


m = D . a . x 

Mass of a stratum 

of air. 


which substituted in the above equation, and writing 












ELEMENTS OF ACOUSTICS. 


29 


therein x for x t in the denominator, from which it does 
not sensibly differ, we have 


_ E x — x t t 

~ I) X X 


Velocity 
imparted to 
stratum CD; 


Now at the end of the time t, the stratum A B has 
reached the position A! B\ and the stratum C D begins 
to move ; that is to say, the disturbance has been propa¬ 
gated over the distance from A to 0 = a?, in the time t. 
Hence, denoting the velocity of this propagation, which is 
that of the wave motion, by V, we have 

> 



t_ _ _ 1 _. 

x ~ V ’ 


this in the last equation gives 

E x — x. 


which may be written 


V. v 


x _ E 
x — x t B 


Molecular 
(2)'. velocity; 

\ 


Here V, is the wave velocity and v, the actual velocity 
of a stratum of air, and for the indefinitely small time t , 
these may be regarded as constant; but the spaces x and 
x — x J are described with these* velocities in the same 
time, and hence 


whence 


x — x t : x : : v : V 


v — 


V . 


x — x t 

- , 


X 


The same in 
other terms; 








so 


NATURAL PHILOSOPHY. 


Wave velocity. 


Same in other 
terms; 


Atmospheric 

density; 


Wave velocity 
for gi ven 
temperature 
and pressure. 


and this substituted above gives 


therefore 




( 3 ) 


whence we see, that the wave velocity in the same medium , 
at a constant temperature and under a constant pressure , 
will he constant , being equal to the square root of the 
ratio obtained by dividing the elastic force of the medium 
by its density. Replacing E by its value as given in 
Eq. ( 2 ), the above reduces to 


y — s/ g-h . jjL .( 4 ). 

§ 21 . The density of the atmosphere or any other 
elastic medium, corresponding to any barometric column 
A, and temperature t , is given by Equation (390) Me¬ 
chanics ; that is, by 

_ — —. _ . 

^ ~ 30“ 1 + (t - 32°) . 0,00208 

and this substituted in equation (4), for D, gives 
V = \/i 7 .30“. E- . |l + ($ - 32°) . 0,00208] . . (5). 

in whicji D n denotes the density of mercury, and D, thal 
of the atmosphere at 32° Fah., the atmosphere being 
under a pressure of 30 inches of mercury. 


Barometric 

height 


§ 22 . The quantity A, does not appear in Equation (5); 
from which we are to infer that the velocity is indepen- 







ELEMENTS OF ACOUSTICS. 


01 

O-l 


dent of the atmospheric pressure, as it should he ; for, an velocity of sound 
increase of pressure will increase the elastic force E ; hut atmospheric ° f 
this will increase the density D, in the same ratio, so that, pressure; 
Equation (3), the velocity should remain unchanged. But 
an increase of temperature under a constant pressure 
dilates the air, and therefore reduces D for the same 
value of E. Hence, all other things being equal, the Velocity s™ 8 ** 7 

7 ° ° x ' in warm weather 

velocity of sound should he greater in warm than in cold than in cold, 
air; greater in summer than in winter, and this is what 
is indicated hy the quantity £, in Equation (5). 

§ 23. If in Equation (5) we make t — 32°, we find 


/ in. T) 

F= V 9 ■ 30 • jf- .( 6 )- 


The density of distilled mercury at 32° Fah. is, Me¬ 
chanics, § 390, equal to 13,598, and that of air at the same 

temperature, and under a pressure of 30 inches = 2.5, of Tabnlarv8hle5 
mercury is 0,001304 ; and the mean value of g is, Media- ^the above 

nics, article 8, equal to 32,1808, which values in Equa¬ 
tion (6) give 


V = 


32,1808 


/ 

2,5 


13,598 

0,0013 


= 915^69 


(fty Velocity of sound 
without increase 
of temperature. 


which would be the velocity of sound in our atmosphere 
under a pressure of 30 inches of mercury and at the tem¬ 
perature of freezing water, were it separated from admix¬ 
ture with all other media. 


§ 24. But it must be remarked that the value of E\ in 
Equation (3), which is one of the important elements of increase of 
this estimate, is assumed to be given by the weight due t 0 tempemture ' ,or 
the height of the mercurial column. How, this only mea- vibration, 
sures the pressure due to the grosser elements of atmo- pr0<luce<lby 

A ° sonorous waves. 

spheric air, and takes no account whatever of the elasticity 







32 


NATURAL PHILOSOPHY. 


Elasticity due to 
the ether. 


Corrected value 
for velocity. 


Velocity as 
affected by 
etherial waves, 
or increase of 
temperature. 
Co-efficient of 
barometric 
elasticity, K. 

To find the 
constant K, V, 
must be known. 


V, affected by 
wind; 


due to that vastly more subtile and refined atmosphere of 
ether which permeates the air, glass, and torricellian 
vacuum, and which, therefore, presses alike on both ends 
of the barometric column. A motion among the atmo¬ 
spheric strata will give rise to a similar motion in this 
ether; the equality in its elasticity on opposite sides of 
the strata in the direction of the motion will be disturbed; 
this inequality will develope a reciprocal action among 
the strata of ether and those of the atmosphere itself; 
hence, E, in Eq. (3), is too small, and consequently F, is 
also too small. 

Denote by X, a constant co-efficient which, when multi¬ 
plied into A, as indicated by the barometer, will give the 
true elastic force as it actually exists; then will Equation 
(5) become 

F=V 7 • 30 [l + (<- 32°). 0,00208] .. (7). 


or, replacing the value of the first three factors as given 
by Equation (6)', 


V = 915,^9 . \/ K . (l + (7-32°) . 0,00208] . .. (7)'. 


The quantity X, may be called the co efficient of baro¬ 
metric elasticity of the air. 

§ 25. To find the value of F, corresponding to any tem¬ 
perature £, it will be first necessary to know that of K. 
But X, being constant, if the value of F be found for 
any particular state of the air, that of X will result from 
equation (7)'. 

The velocity V, is the rate of travel of the front of the 
wave from a disturbed particle of air taken as an origin. 
When the wind blows, the whole mass of air, and there- 





ELEMENTS OF ACOUSTICS. 


33 


fore this origin, has a motion of translation ; and to find To find v 
V experimentally, the observations should be so eon- ex P enmontaIt7; 
ducted as to eliminate the disturbing effect of the wind. 

To understand how this may be 
done, suppose an observer placed Fls ' 16 ' 

at A, midway between two sta¬ 
tions B and (7, and the wind to % - 2 - —c 

blow from B to G. Denote the 

velocity of the wind by v ; then will the velocity with 
which sound will travel from B to A, be V + v, and 
from G to A, it will be V — v, the mean of which is 
obviously V. 

To eliminate therefore the effect of the wind, let four 
remote stations B , (7, 7), 77, be so chosen that the line 
connecting G and 7>, shall be perpendicular, or nearly so, 
to that joining E and 7>, and place an observer at the inter¬ 
section A. At the stations B, 77, (7, E\ let signal guns 
be fired in succession, and the observer at A note, by a 
stop watch, the intervals of time between his seeing the 
flash and hearing the report. The distances from A, 
being carefully measured and each divided by the corres¬ 
ponding interval in seconds, will give a value for V. 

The mean of these values and the reading of the thermome¬ 
ter, which must also be noted, being substituted in Eq. 

(7)', the value of K will result. 

The experiments of Moll, Vanbeek and Kuyten- Experiment 
brouwer, performed in 1823, over-a distance of 57839 
feet, in a dry atmosphere, at the temperature of 32° Fahr., 
gave a mean value of V= 1089,42 English feet. These 
values substituted in Equation (7)' give 




which in Eq. (7)' gives the general value of 


V = 1089J42 v/1 +{t- 32°). 0,00208 


Final value of 
the velocity V. * 







34 


NATURAL PHILOSOPHY. 


Principle of beat. §26. This vibratory motion among the elements of 

ether, giving rise to a secondary system of waves, by 
which the propagation of sound is accelerated, constitutes 
the principle of heat. And to ascertain to what degree a 
Fahrenheit thermometer would be affected were it sud¬ 
denly transferred from a perfectly stagnant atmosphere to 
one agitated by sound waves, could the mercury take 
instantaneously the bulk which would enable its ether to 
vibrate in unison with that of the sound wave, it would 
only be necessary to find the value of t — 32°, in Equation 

(5), after substituting for V and fJ . hlhj, their respec- 

tive values 1089,42 and 915,69. Solving the equation 
with reference to t— 32°, and introducing these values, 
we find, 


Amount of latent 
heat rendered 
free. 


-32°= —_[ / 1 0 . 8 ?’ 42 ) 3 - 1 1 = 199,71. 

0,00208 L \ 915,69 / I 


Difference 

between 

computed and 

observed 

volocity 

explained. 


This is called the amount of heat given out by an element 
of air during its condensation in a sound wave. It was to 
the increased elasticity imparted to air by this sudden 
change of a portion of its heat from latent to free , that 
Laplace first attributed the great disparity between the 
computed and observed velocity of sound. 


Effect on the 
stratum CD 
resumed. 


Two cases may 
arise; 


Fig. 15. 

C JE 


§ 27. Before proceeding further we must remark, that 
nothing has been said of the conduct 
of the stratum C D , after it was im¬ 
pelled forward from its place of rela¬ 
tive rest by the action of the stratum 
A j&, which was brought by the 
disturbing cause, say the motion of 
a rigid plane, to the position A' B r . 

Two cases may occur : either the stratum A B may be 
retained in the position A'B\ or the disturbing piano 
may, by an opposite movement, leave this stratum unsup¬ 
ported from behind. In the first case, if the medium bo 


First Cflso; 











ELEMENTS OF ACOUSTICS. 


35 


homogeneous, the masses of all its particles will be equal, in first case a 
and the velocity impressed upon those in the stratum ^ansm^tted in 
CD will, by the principle of the collision of elastic masses, the direction of 
be transferred undiminished to those in the stratum E F , the dlsfcurbauw: 
after which the stratum C B will come to rest; and the 
same of the succeeding strata in front: Mechanics, § 209; 
so that there will simply be a pulse, transmitted along the 
direction in which the primitive disturbance acted. In 
the second case, the stratum A' B\ being left unsupported in the second 
from behind, by reason of rarefaction, will be thrust back- 
ward by the superior elasticity of the medium in front, transmitted in 
and this return or backward motion will take place in all 
the strata in front, in the same order of time and distance 
from the original disturbance as in the instance of the 
forward movement; so that a second pulse will be trans¬ 
mitted in the same direction as before, only differing from 
the first in the backward motion among the parti¬ 
cles. 


Distances 

§ 28. It is easy from the known velocity of sound, to 
compute the distance between two places which may be sound; 
seen, the one from the other; and for this purpose let a 
gun be fired at one place, and the interval of time between 
seeing the flash and hearing the report at the other be 
carefully noted. This interval, expressed in seconds, mul¬ 
tiplied by 1089,42 y/ 1 + (t — 32°). 0,00208, will give the 
distance expressed in English feet. The value of t will 
be given by the Fahr. thermometer. Accuracysiigbtiy 

The accuracy of this determination will of course be affectedby wind * 
affected by the wind, should it be blowing at the time. 

To ascertain the probable amount 

of this influence, let A be a sta- Flg * m 

tion midway between the places 

B and O, and suppose the wind - -—- c 

to be blowing from B to (7, with 

a velocity denoted by vf denote the distance B A — G A determined . 
by 8, then will the actual velocity of sound from B to A , 
be V+v, and from C to A, be F— v; and the intervals 


Influence of w ind 





36 


NATURAL PHILOSOPHY. 


Intervals of time 


First interval; 


Second interval; 


True interval; 


Resulting 
formula for 
distance. 

Example 


Distance from 
Wwt Point to 
Newburgh. 


of time observed at A, between the flash and report from 
B and (7, will be, respectively, 


8 

V + v 



or developing these expressions, 



Now, the most violent hurricane moves at a rate less 
than one-tenth that of sound; so that the neglect of the 
terms involving v 2 , would in the worst case only involve 
an error less than ^ ¥ th, and in the ordinary cases likely 
to be selected for experiment their influence would be 
quite inappreciable. Neglecting these terms, we see that 
one of these intervals will be just as much too great as 
the other is too small, and the true interval, denoted by t , 
will be a mean between them. Hence, 

^ _ t x + t 2 _ S 
1 - 2 “ T~’ 

or 

S = V.t .(9). 

Example. On the occasion of firing a salute of 13 
minute guns at Newburgh, the mean of the intervals be¬ 
tween noting the flash of each gun and hearing the 
report at West Point, N. Y., was 36,2 seconds; and the 
temperature of the air, as given by a Fahr. thermometer, 
was 76° ; required the distance from West Point to New¬ 
burgh. 

S=t. V= 36,2.1089^42 V 1 + (76 a - 32°). 0,00208 








ELEMENTS OF ACOUSTICS. 


37 


S- 36*2.1089,42 V 1,0915 
and by logarithms: 


36,2 1,5587086 Computation. 

1089,42 . 3,0371954 

1,0915, (i), .... 0,0 190118 

41,202 feet.4,6149158 

5280, feet in 1 mile, ac. . . 6,2773661 
7,8034 miles, . . . 0,8922819 


. § 29. We have seen that the velocity of sound through Veloclty 
the air is independent of the barometric pressure, and independent of 
experiments show it to be sensibly unaffected by its ^teonhe ^* 1 
hygrometrical state of moisture and dryness; the actual atmosphere, 
weather characterised by fog, rain, snow, sunshine; the ^umiTa' 6 
nature of the sound itself, whether produced by a blow, 
gunshot, the voice or musical instrument; the original 
direction of the sound, whether the muzzle of the gun is 
turned one way or the other; the nature and position of 
the ground over which the sound is conveyed, whether 
smooth or rough, horizontal or sloping, moist or dry. 

§ 30. Resuming Eq. (7), and denoting by V' and p™ velocity of sound 
the velocities of sound through any two gases whatever, 
by K' and K" their co-efficients of barometric elasticity, 
and by D’ and D” their densities; then, supposing the 
barometric column exposed to the pressures of the gases 
to be 30 inches, and the temperature of the gases 
to be the same and equal to t degrees, will, Eq. (7), give 

V’ = 0“-j£7iT 1 + {t - 32°) . 0,00208j Value to first, 

and 


F"= \/<j • ^"[l + (t - 32°). 0,00208j ; 


Value in second; 











38 


NATURAL PHILOSOPHY. 


Velocities 

compared. 


Cotaclueion. 


Atmospheric air 
and hydrogen; 


Iiatio of their 
constant 
coefficients; 


Inference; 


Conforms t® 
Boscovich’s 
theory. 

f 


Experiments on 
liquids. 


Dividing the first by the second, we have 


XL _ /EL XXL 

v" ~ * k " * jy 

That is, the velocities of sound in any two gases, at the 
same temperature, are to each other as the square roots of 
their coefficients of barometric elasticities directly, and 
densities inversely. 

From Equation (10) we readily obtain 


K' _ V^_ D' 

K" ~ y>'2 * d" 

Taking one of the gases atmospheric air, and the other 
hydrogen, and assuming the velocity of sound in hydro¬ 
gen, as determined by the experiments of Yan Rees, Fra- 
meyer and Moll, to wit, 2999,4 English feet, we have, 
after substituting the known values of the quantities in 
the second member, 


K* 

K" 


/ 2999,4 0 068g _ 0 5215 

\ 1089,42/ ’ ’ 


Hence the coefficient of barometric elasticity of air is 
nearly double that of hydrogen; a result which appears 
to indicate that the velocity with which sound is propa¬ 
gated through gases is in some way dependent upon their 
chemical or physical constitution. This would seem but 
the natural consequence of the views of Boscovich. 


VELOCITY OF SOUND IN LIQUIDS. 


§ 31. From the experiments of Canton, Oersted, and 
others, liquids as well as gases are found to be both com- 




ELEMENTS OF ACOUSTICS. 


39 


pressible and elastic ; and are therefore fit media for the Experiments on 
transmission of sound. From the experiments of Colla- pure water; 
don and Sturm, on what may be regarded as pure water, 

Sir John Herschel deduces the compression of this fluid, 
by one standard atmosphere, to be 0,000049589 =e; that 
is to say, an increase of pressure equal to that arising from 
a column of mercury having an altitude of 30 inches and 
temperature of 32° Fahr., will produce a diminution in 
the bulk of water equal to -■ 0 0 4 o y o 8 <> ' o 0 o °f the entire Deductl0n - 
volume which it had before this increase. 


§ 32. All bodies may be stretched or compressed by the 
application of force, and when unaccompanied by perma- Law of dl6(< 
nent change of molecular arrangement, the degree of com- tion. 
pression or extension is directly proportional to the intensity 
of the force which produces it. 


§ 33. Denote by M and B, the intensities of two forces 
capable of stretching a body, whose cross-section is equal 
to unity, to double its natural length ,L ) and to L + 1, 
respectively; then will 

L : l:: M: B; B = M. \ =M. e, 

JJ 


Measure of 
elastic force. 


m which M is called the coefficient or modulus of elasticity. 


Fig. 15. 


§ 34. Let AB , and CD , be two consecutive strata of 
water, and suppose the stratum AB , to 
have been suddenly moved by some 
disturbing cause to the position A! B'. 

Denote the distance BD by a?, and 
B'Dhj x ,, then, regarding the area 
of the stratum as unity, will the dif¬ 
ference of volume between ABCD and A'B’ CD, be 
represented by x — a? y , and the degree of compression 
referred to the original volume, by 


Illustration; 


X — x t 


X 


Degree of 
compression; 











40 


NATURAL PHILOSOPHY. 


Compressing 

torce\. 


its value. 


Combined 
pressure on a 
stratum below 
the surface; 


Moving force on 
a stratum; 


Mass of a 
stratum; 


Velocity 
generated in a 
unit of time; 


Velocity in an 
elementary 
portion of time. 


and the force necessary to produce this compression 
will, § 4, be given by the proportion 


Me : M. 



B : E. 


in which B = g . D u . A, denotes the pressure due to a stan¬ 
dard atmosphere, being the weight of a column of mer¬ 
cury whose density is D n and height A. Whence 


E, 


B x — x t 
e * x 


But any stratum of water situated below the surface 
is already subjected to the pressure of the atmosphere, 
and that arising from the weight of the column of the 
same fluid above it. Denoting this combined pressure 
by p, we shall have the stratum A' B\ and therefore 
C B, since the resistance to compression arises from the 
reaction of the latter, urged forward toward EE, by 
E t + jp ; but the motion of C B is resisted by the pres¬ 
sure whence the moving force becomes ^ +%> — p = E t . 
The mass of the stratum CD will, § 20, be 

B. x 

whence the acceleration due to the moving force, or the 
velocity generated in a unit of time, becomes, after substi-. 
tution for B, its value, 


E t _g-h-D u 1 x — x, 
Dx ~ e • D x x 


and the velocity v, imparted to the stratum 0 B, in an 
elementary portion of time £, will be given, Mechanics, 
Eq. (23), by the equation 





ELEMENTS OF ACOUSTICS. 


41 


v 


9-h-D „ . I , x — x, 
e • D x x 


Its value; 


but, § 20, 



which substituted above gives, §33, after clearing the 
fraction and extracting the square root, 


V = \ / 

v e.D 



Wave ve!6«ity 

(12) watef 5 


and substituting the numerical values of < 7 = 32 , 1808 ; h = 

30* n - = 2,5 ; D u — 13,598 ; and denoting by iT, what we Numerical value* 
have before termed the co-efficient of barometric elasti- of data; 
city, we finally have 



32,1808 . 2,5 . 13,598 
0,000049589 . D 


^=4696,86 


K 

D 


Resulting wave 

(13) velocity; 


in which D must be taken from the table given in §276, 
Mechanics, corresponding to the temperature of the 
water. If the temperature of the water be 38°,75 Fahi\ 
Z> will be unity, and if we assume K — 1, then will 


V= 4696,86. 


Velocity when 
density and 
constant 

§ 35. A careful and doubtless most exact experimental 
determination of the velocity of sound in water was made unity - 
in 1826, by M. Colladon. After trying various means for Experiments of 
the production of sound under water, he adopted the bell, Coiiadon; 
as giving the most instantaneous and intense sound, the 
blow being struck about a yard below the surface by 
means of a metallic lever. The experiments were made 











42 


NATURAL PHILOSOPHY. 


Explanation; 


Mean of 
Intervals; 


V locity of 
sound in water; 


Four times as 
great as in air. 


Experimental 
determination of 
the constant K 
for water; 


at night, the better to avoid the interference from extra¬ 
neous sounds, and to enable him to see the flash of gun¬ 
powder which was fired simultaneously with the blow. 
To receive the sound from the water and convey it to the 
ear, a thin cylinder of tin, about three yards long and 
eight inches in diameter, was plunged vertically into the 
water, the lower end being closed and the upper end, to 
which the ear was applied, open to the air. By means of 
this arrangement he was enabled to hear the strokes of the 
bell under water across the entire width of the Lake of 
Geneva from Rolle to Thonon, a distance of about nine 
miles. 


§ 36. From 44 observations, made on three different 
days, it appears that the distance of 44249,3 feet was 
traversed in 9,4 seconds, this being the mean of the 
intervals between the instant of seeing the flash and 
receiving the sound at the cylinder, the greatest deviation 
from which of any single observation not exceeding 
three-tenths of a second; which gives 


V = 4 i|^ = 4707,4 .... (14) 


thus making the velocity of sound in water more than 
four times as great as in air. 


§ 37. The mean temperature of the water, taken at 
both stations and midway between them, was 46°,6 Fahr. 
and its specific gravity was found to be exactly that of 
distilled water at its maximum density, viz.: unity, the 
expansion arising from the excess of temperature being 
just counterbalanced by the superior density due to the 
saline contents. This circumstance furnishes at once the 
means of finding the numerical value of the coefficient K; 
for by making D =1, in equation (13), and equating the 
resulting value of F, and that given above, we have 




ELEMENTS OF ACOUSTICS. 


43 



1,00449, 


Value of K; 


which differs but little from unity, and from which we 
infer that there is but little heat developed in the trans¬ 
mission of sound through water. And the experiments Inference with 

° regard to liquids, 

hitherto made indicate that this is also true of other 
liquids. To find the velocity of sound in any liquid it 
will only be necessary to know its compressibility. A 
valuable table of the compressibility of different liquids 
is given by Sir John Herschel, in his Treatise on Sound, 

Encyc. Met., Yol. 1, p. 770. 


§ 38. In these experiments of M. Colladon, it was found Different tones 
that the sound of the bell when struck under water if of8oandin 

water and air. 

heard at a distance had no resemblance to its sound m 
air. Instead of a continued tone, a short sharp sound 
was heard like two knife blades struck together; it was 
only within the distance of about six hundred yards that 
the tone of the bell could be distinguished. 

§ 39. M. Colladon also found that sound in water does Sound in water 
not, like sound in the air, spread round the corners 0 f notaudib,e 

' 1 around corners 

interposed obstacles. In air, a listener situated behind a as in air. 
projecting wall or corner of a building, hears distinctly, 
and often with very little diminution of intensity, sounds 
excited beyond it. But in water this was far from being 
the case. When the tin cylinder, or hearing tube, before 
mentioned, was plunged into water at a place screened 
from rectilinear communication with the bell, by a wall 
running out from the shore, and whose top rose above the 
water, a very remarkable diminution of intensity was 
heard in comparison with that observed at a point equally 
distant frofn but in direct communication with, the bell, 
or “ out of the acoustic shadow .” 

The reason of this apparently singular phenomenon will Acoustlc shadow * 
appear further on. 




44 


NATURAL PHILOSOPHY. 


Solida propagate 
sound better 
than gases or 
liquids; 


Same formula 
employed; 


Difference in 
structure of 
solids and 
liquids; 


Consequences of 
this difference; 


Solids differ from 
©ach other in 
molecular 
arrangement; 


Effect upon the 
velocity of 
sound. 


VELOCITY OF SOUND IN SOLIDS. 

§ 40. Solids, when elastic, are even better adapted to 
the transmission of sounds than gases or liquids. But for 
this purpose, they should be homogeneous in substance, 
and uniform in structure. The general principle upon 
which the propagation of sound through solids depends is 
the same as in liquids; and the same formula, Eq. (12), 
may be employed when the intensity of the specific 
elastic force Me, § 33, of the solid is known. There are, 
however, two very important particulars in which they 
differ. First, the molecules of liquids admit of a perma¬ 
nent change of relative position among themselves; those 
of a solid are, on the other hand, as before remarked, 
§ 16, subjected to the condition of never permanently 
altering their relative arrangements without altering their 
physical character. Second, each particle of a liquid is 
similarly related to those around it in all directions; 
while every particle of a solid has distinct sides and 
different relations to space and surrounding particles.^ 
Hence arise a multitude of qualifying circumstances, 
which modify the propagation of sonorous waves through 
solids, which have no place in liquids, and peculiarities ot 
wave motion become, therefore, possible in the former 
which are impossible in the latter. 

§ 41. Solids differ much among themselves in the 
particulars here referred to. Thus, the cohesion of the 
particles of crystallised bodies differs greatly on their 
different sides, as the facility with which they admit 
of cleavage in some directions and not in others, shows. 
They have different elastic forces in different tdirections, 
and thus the velocity of sound through them must de¬ 
pend, Eq. (12), upon the direction in which the sound is 
transmitted. A disturbed particle in a perfectly homo- 



ELEMENTS OF ACOUSTICS. 


45 


geneous medium becomes the centre of a series of con¬ 
centric spherical waves which proceed outwards with 
equal velocities in all directions. But if the elastic force Medium not 
and density of the medium vary in different directions hom ° 6e eo 
from the place of disturbance, Equation (12), shows that 
the shape of the wave front will no longer be spherical. 

§ 42. The most general hypothesis with regard to the 
constitution of solids, is that which attributes a different 
elasticity in three directions at right angles to one another; 
and if these elasticities may be measured by three lines, 
drawn from a common origin in these directions, and whose 
lengths are denoted by a, b and c, respectively, and r meas¬ 
ure the elasticity in any other direction, then will 


r 2 = a 2 . cos 2 a + b 2 cos 2 /3 -f c 2 . cos 2 y, 

in which a, j3 and y denote the angles which r makes with 
a, b and c, respectively. (Analytical Mechanics, § 318.) 
The surface of which this is the equation is called, the sur¬ 
face of elasticity, and the lines a, b and c, are called axes 
of elasticity. 

In a solid thus constituted, the wave shape will be given 
* by the equation, 


(x 2 + y 2 + z 2 ) ( a 2 x 2 + b 2 y 2 -\- c 2 z 2 )" 


— a 2 (b 2 + c 2 ) x‘ 

— b 2 (a 2 + c 2 ) y 2 

— c 2 {a 2 -f b 2 ) z 2 
+ a 2 b 2 t 


^ = 0 


General wave 
surface. 


• ( 16 ) 


in which x, y, z are the co-ordinates of any point of the 
wave surface. (Analytical Mechanics, § 319.) 

If' the elasticity in the direction of two of the axes be 
equal, that is, if b = c, then will Eq. (16) become 

Particular case* 


(x 2 -f- y 2 + z- — c 2 ) [a 2 x 2 + c 2 {y 2 + z 2 ) — a 2 c 2 ] =0 . (17) 




46 


NATURAL PHILOSOPHY. 


Wave resolution. 


Consequences of 
wave resolution. 


Different wave 
velocities. 


Velocity of 
6ound in various 
solids. 


Duplication of 
sound. 


and the wave resolves itself into two distinct waves, the 
one a sphere, of which the radius is c, and the other an 
ellipsoid, of which the semi-axes are a and c. 

And thus, a wave of sound entering such a body from 
the air or other homogeneous medium, would separate into 
two components which would travel with different velo¬ 
cities in every direction except that of the axis a. This 
difference of velocity would vary with the inclination of 
the wave motion to the same axis, aud the distance by 
which one component would lag behind the other on any 
given line through the centre, would be measured by the 
difference between the radius vector of the ellipsoid and 
radius of the sphere coincident in direction with this line. 
(See § 142, Optics.) 

Upon these facts depend some of the most curious and 
important phenomena of optics. 

§ 43. By a series of experiments similar in principle to 
those already referred to, and which it is unnecessary to 
detail, it is found that the following are the velocities of 
sound in different solids, that in air being taken as unity, 
viz.: Tin = T \ ; Silver = 9 ; Copper = 12 ; Iron (steel ?) 
= IT ; Glass = IT ; Baked Clay (porcelain?) = 10 to 12; 
Woods of various species = 11 to IT. 

It was found by IIerhold and Rafr, that when a metal¬ 
lic wire 600 feet long, stretched horizontally and held at one 
end between the teeth, was struck at the other, two dis¬ 
tinct sounds were heard-; the one transmitted through the 
wire, teeth and solid materials of the head, to the audi¬ 
tory nerves, the other through the air. A similar dupli¬ 
cation of sound was observed by ITassenfrats and Gay 
Lussac from a blow struck with a hammer against the 
solid rocks in the quarries of Paris; that propagated 
through the rock arriving almost instantly, while that 
transmitted by the air lagged behind. 


§ 44. From this it is easy to estimate the time required 



ELEMENTS OF ACOUSTICS. 


47 


to transmit the effect of a force applied at one end of a Time re( i uirC(1 ^ 
solid, or arrangement of solids, to the other. In iron, for effect of a force, 
instance, the effect of a push, pull or blow, will be propa¬ 
gated towards its point of action at the rate of 11090 feet 
a second after its first emanation from the motor. For all Examples; 
^moderate distances, therefore, the interval is utterly in¬ 
sensible. But Sir John Heeschel remarks, that if the sun 
were connected with the earth by an iron bar, no less 
than 1074 days, or nearly three years, must elapse before sun and earth, 
the effect of a force applied at the former body could 
reach the latter. Yet the force actually exerted by the 
mutual gravity of the sun and earth may be proved to 
require no appreciable time for its transmission. 


PITCH, INTENSITY AND QUALITY OF SOUND. 

§ 45. We have seen that the velocity of sound, in the velocity constant 
same homogeneous medium, is constant; and that the !. n a 
particles in any one wave, or set of waves, arising from the medium; 
same disturbance, all perform their revolutions in equal Wave 1( T st !' , 
times. And hence, Equation (1), the waves flowing from the position : 
same agitating cause are of the same length, no matter to 
what distance they may have been transmitted. This 
length of wave, Equation (1), varies directly as the time 
of revolution of a single particle. In proportion as this 
time is shorter, so will the wave be shorter, and in propor¬ 
tion as it is longer, will the wave be longer. And since But varies 
the particles of the auditory nerves vibrate in harmony 88 the 
with those of the waves which agitate them, the number revolution of a 
of recurrences of the same condition of these nerves, in a partlcle ‘ 
given time, will depend upon the length of the waves. 

The greater or less number of these recurrences deter¬ 
mines the character of the sound; in proportion as this 
number is greater will the sound be less grave or more Acute and 
acute, and in proportion as it is less, will the sound be grave sounds, 
less acute or more grave. This particular character of 



48 


NATURAL PHILOSOPHY. 


Pitch. 

Time of 
revolution of a 
particle depends 
on disturbing 
cause. 


Short and long 


High and low 
n< tes. 


sound by which it is pronounced to be grave or acute, 
more grave or more acute, is called the Pitch. 

The time of vibration of a single particle in any wave 
depends, §18, upon the disturbing cause. The waves 
projected through the air by the sluggish vibrations of 
the coarse and heavy strings of the largest violins, called 
“double bass” are, therefore, long, and the correspond¬ 
ing sound is grave; while the waves produced by the 
more rapid vibrations of the fine and tense strings of the 
violin proper, are shorter, and the sound is acute. In the 
latter case the pitch is high; in the former low ; and hence 
the terms high and low notes in musical instruments. 


Wave length 
independent of 
excursions of 
particles; 


Waves may be 
equal in length 
while the 
particles have 
different 
velocities; 


Quantity of 
action in particles 
different; 


Intensity or 
loudness; 


Examples, violin, 
piano. 


§ 46. A wave being once excited, the time of vibration 
of any one of its particles, and therefore the length of 
the wave itself, becomes wholly independent of the dis¬ 
tance to which the particle may recede from its place of 
relative rest. But, in order that the time may not vary, 
those particles must move at the greatest rate which 
make the greatest excursions. Hence, theVe may exist 
many waves of the same length while the particles of 
one possess very different velocities from those of 
another. The quantity of action in each particle being 
equal to half of its living force, or equal to half the pro¬ 
duct of its mass by the square of its velocity, the par¬ 
ticles of air in these different waves will assail the audi¬ 
tory nerves with very different efforts; and this it is 
which constitutes the distinction we observe between tw T o 
sounds of the same pitch possessing different degrees of in¬ 
tensity, or, as it is usually expressed, different degrees of 
loudness. Thus, when the string of a violin or of a piano is 
drawn aside and abandoned to itself, it will vibrate about 
its position of equilibrium for some time, and finally come 
to rest. The sound, which at first is loud, gradually dies 
away, and ultimately ceases. But we only hear one con¬ 
stant pitch as long as the string moves bodily to and fro. 
It is easily shown that the time of each vibration of the 
string is the same from the beginning to the end of the 



ELEMENTS OF ACOUSTICS. 


49 


motion; the lengths of the sonorous waves impressed Constaucy of 

upon the air must, therefore, be invariable, and hence 

the constancy of pitch. On the contrary, the distance by 

which the string departs from its place of rest in each 

vibration, gradually diminishes, and so does that of the 

aerial particles, whose motions are regulated by those of Gra<Jual deca y of 

the string; this explains the gradual decay of the 

sound. 

§ 47. Sounds may have the same pitch and intensity 
and yet be very different. We never confound, for ex¬ 
ample, the sound of a trumpet with that of a violin, not- Quality of sound; 
withstanding these sounds may have the same degree 
of acuteness and loudness. And this fact gives rise to a 
distinction of quality. 

Thus far nothing has been said of the peculiarities 
which mark the mode of vibration of the elements of 
a sonorous wave; whether, for instance, the particles 
describe elliptical, circular, or rectilinear orbits ; whether 
the planes of these orbits are perpendicular, inclined, or 
parallel to the direction of the wave propagation. Nor 
has it been necessary to discuss these particulars, since 
the velocity, pitch and intensity are wholly independent 
of these considerations. But while the amplitude and 
time of vibration of the particles of the auditory nerves, 
induced by different sonorous waves, may be the same, Determined by 
thus inducing a constant intensity and pitch, yet the C0r "^ e ^^ yof 
responding sensations may derive a .peculiarity of hue, so 
to speak, from the variations in the mode of molecular 
motions above referred to, sufficient to account for the 
distinction of quality. 

S 48. To ascertain what length of wave corresponds to Lon s th of wav0 

. ... , . corresponding to 

our sensation of a particular pitch, we must have the a certain pitch 
means of measuring the lengths of different waves. These determined by 
are furnished in an elegant little instrument called the 
Siren ; a device of Baron Cogniaed de la Toue. In 
this instrument the wind of a bellows is emitted through 

4 



50 


NATURAL PHILOSOPHY. 


Fig. 18. 



simi; a small hollow tube A, 

before the end of which 
a circular disc B, pierc¬ 
ed with a number of 
equal and equidistant 
holes arranged in the 
circumference of a cir¬ 
cle concentric with the 
axis of motion 67, is 
made to revolve. The 

tube through which the air passes is so situated that the 
holes in the disc shall pass in rapid succession over its 
open end and permit the air to escape, being at the same 
time so near to the plane of the disc that intervals be¬ 
tween the holes serve as a cover to intercejjt the air. If 
construction and the holes be pierced obliquely, the action of the current 
of air alone will be sufficient to put the disc in motion; 
if perpendicular to the surface it must be moved by 
wheel work, so contrived as to accelerate or retard the 
rotation at pleasure. The bellows bein'g inflated and 
the disc put in motion, a series of rapid impulses are 
communicated to the air in front of the holes; and, 
when the rotation is sufficiently rapid, a musical tone is 
produced whose pitch becomes more acute in proportion 
as the velocity of rotation increases. To show that the 
air of the bellows only acts as a mass in motion to im¬ 
press by its living force successive blows upon the ex¬ 
ternal air, the bellows may be replaced by a reservoir 
of water, the liquid being under sufficient head to cause 
it to spout through the holes of the disc as they come 
successively in front of the duct pipe; the effect is the 
same. 

Connected with the axis of rotation of the disc are a 
stop-register, which indicates the number of revolutions, 
and a stop-watch, to mark the time in which these revo- 
siop-regietcr and lntions are actually performed. The instrument being 
put in motion and accelerated to the desired pitch, the 
register and watch are relieved from the stops, and after 


Bellows uiay be 
replaced by a 
reservoir of 
water: 


atop-watch; 



ELEMENTS OF ACOUSTICS. 


51 


the sound has continued for any desired length of time, Reading of the 
the stops are again interposed, and a simple inspection of 
the dial plates of the watch and register will give the 
time and number of revolutions. 

[Now, suppose the disc to be pierced with m holes, the 
number of revolutions to be n , and the number of seconds 
to be T. The number of impulses, and therefore the 
number of waves, will be m . n ; and the number of waves 
produced in one second will be 

Number of waves 
in one second; 

But these waves, generated in one second, occupy the 
entire distance denoted by V, the velocity of sound; and 
hence, denoting by A, the wave length, we have the 
relation, Equation (8), 


m . n 
~~T~ * 


. /I = V = 1089,42 . v' 1 + (;( - 32°) . 0,00208. Fonm.!*; 
whence, 


1089,42 . T. Vl+{t - 32°) 0,00208 /1Q , Value for wave 

A = ——-— • • (I®), length. 

m . n 

Example. Suppose the revolving disc to be pierced with Example; 
100 holes, the time of rotation 20 seconds, the number of 
revolutions in this time 102,4, and the temperature of the 
air 84° Fahr. Then will 

m = 100; n = 102,4; T= 20*-; t = 84°, 
which in Equation (18), give 




1089,42.20 . v/ 1+52.0,00208 * 

100.102,4 “ 5 


Vttluo oi X 










52 


NATURAL PHILOSOPHY 


Results of thus making the length of the wave two and $ quarter 

experiments; English f ee t, nearly. 

The results of the experimental researches of M. Biot, 
on this subject, are given in the following table : 


mie. 


Number of vibrations 

Length of resulting ware 


in one second. 

in English feet 


1. 

. . . 1091,34 


2. 

. . . 545,67 


4. 

. . . 272,83 


, 32. 

.... 34,10 


( 64. 

. . . 17,05 

1 1 

128. 

.... 8,52 

!«• 

\ 256 . 

.... 4,26 

i i, 

/ 512. 

. . . . 2,13 

11 

jl024 . 

.... 1,06 

0 5 

/ 2048 . 

.... 0,53 

1 2 1 
.0 

1 4096 . 

.... 0,26 

1 

\ 8192. 

.... 0,13 


Lowest audible § 49. From these experiments it has been inferred that 
pitch. the lowest pitch audible to the human ear, is that pro¬ 

duced by a wave whose length is 34,10 English feet, and 
of which there are generated, in one second of time, 32 in 
Highest audible number ; and that the highest audible pitch is given by a 
wave whose length is 0,13 of an English foot, or about 
one and a half English inches, and of which 8192 are 
generated in a second. But in such experiments much 
must depend upon the ear of the experimenter ; we know 
that this organ differs greatly in different persons, even 
among those who are unconscious of any defect in their 
sense of hearing. Some have contended for a high pro¬ 
bability that a body making 24000 vibrations in one 
second, produces a sound which, to a fine ear, is distinctly 
audible; and M. Savart, by means of a rotary cog-wheel, 
Results vary with so arranged that each tooth should strike a piece of card, 
experimenters, found that 12000 strokes on the card in one second, pro- 















ELEMENTS OF ACOUSTICS. 


53 


duced a sound perfectly audible, as a musical tone of high Powers of the 
pitch. Although different authorities differ in regard earlimited » 
the powers of the ear, they nevertheless all agree in 
ascribing to them a limit. And thus, of the almost end¬ 
less variety of waves which must, from the existence of 
ceaseless sources of disturbance, pervade the air, our 
organs of hearing appear to excite the mind to impres¬ 
sions of those only whose lengths range within certain 
prescribed limits. Nor is this limitation peculiar to the 
ear. We shall have occasion, when speaking of light, to 
remark the same thing of the eye. We shall find that same true forth® 
when, from too small or too great lengths, the waves of eye ’ 
ether lose the power of stimulating the optic nerve to the 
sensation of light, they nevertheless do, when addressed 
to other organs, give rise to the further and obvious sen¬ 
sations of heat. And to what extent we are uncon- our senses cannot 
sciously influenced by those agitations of surrounding al1 

media which fall beyond the range of the ordinary senses 
to appreciate, it would be out of place here to inquire. 

§ 50. There is nothing in the constitution of the The sensations of 
atmosphere to prevent the existence of wave pulses p° r ^iy 
incomparably shorter and more rapid than those of which where ours end; 
we are conscious ; and we are justified in the belief that 
there are animals whose powers in this respect begin 
where ours end, and which may have the faculty of hear¬ 
ing sounds of a much higher pitch than any we actually 
know from experience to exist. And it is not improba¬ 
ble that there are insects endued with a power to excite, 
and a sense to perceive, vibrations of the same nature as 
those which constitute our ordinary sounds, yet of wave 
dimensions so different, that the animal which perceives such animals 
them may be said to possess a different sense, agreeing may 841310 

J 1 . . . 7 . ° possess a 

with our own in the medium by which it is excited, yet different senea. 
entirely unaffected by those slower and longer vibrations 
of which we are sensible. 



54 


NATURAL PHILOSOPHY. 


DIVERGENCE AND DECAY OF SOUND. 


^hlricaiin 78 § ^1. We have already stated, § 16, that when the pri- 
homogenoous mitive agitation of a medium is confined to a small space, 
the initial wave front is of a spherical shape, and we have 
seen, Equation (12), that the sound wave proceeds with 
equal velocity in all directions in which the density and 
elastic force are the same. In all homogeneous media, 
the wave front will, therefore, retain its sphericity to 
whatever distance it may he propagated, and a sound 
equallytaaU Card P r0( ^ UCe ^ a gi yen point, as from the blow of a ham- 
directions; mer, or the explosion of gunpowder, will be heard equally 
well in all directions. 


Not true when 
density and 
elastic force 
vary; 


Illustrated by 
toning fork; 


§ 52. When, however, sounds proceed from a series of 
points situated upon the surface or face of a solid, the 
body of which interposes to prevent the' existence of 
equal density and, elastic force in all directions from the 
points of disturbance, this equality of transmission in all 
directions no longer obtains. This is well illustrated by 
an experiment due to Dr. Young. A common tuning 
fork, a piece of steel, whose shape is repre¬ 
sented in the figure, being struck sharply 
and held with its handle A. against some 
hard substance, is thrown into a state of 
vibration, its branches B , alternately ap¬ 

proaching to and receding from each other. 

Each branch sets the particles of air in 
motion, and a sound of a certain pitch is 
produced. But this sound is very unequally 
audible in different directions. When held 
with its axis of symmetry vertical and at the 
distance of about a foot from the ear, and 
turned gradually about this axis, .it is found 
that at every quarter of a revolution, the 


Fig. 19. 











ELEMENTS OF ACOUSTICS. 


55 


sound becomes so faint as scarce¬ 
ly to be heard, the audible posi¬ 
tions of the ear being in planes 
EE and 0 0 , perpendicular 
and parallel to the broader faces 
of the fork; the inaudible, in 
planes E' E' and O' 0\ mak¬ 
ing with the first, angles of 
45°. 


Audible and 
20 * inaudible 

q positions of tho 



§ 53. To resume the consideration of sound propagated Loudness of 
from a central point. The intensity or loudness of sound sound 
is, §46, determined by the living force with which the determined * 
particles of a medium in sensible contact with the ear 
act upon the auditory nerves. At the primitive point Noloasoflivillg 
of disturbance the living force is impressed by the dis- force in elastic 
turbing cause, and is transferred from the particles of media; 
one wave to those of another without loss, provided the 
molecular arrangements of the medium in the process 
are not permanently altered, Mechanics, § 210, which is 
the case in all elastic media, such as the air and other 
gases when not confined. The sum of the living forces gumofllvln<r 
of the particles in a wave must, therefore, be constant, forces of particiea 
to whatever distance the wave be propagated, and equal 
to double the quantity of work expended by the dis¬ 
turbing motor. The living force of any single particle 
is equal to the product of its mass into the square of 
its velocity, and from the nature of the wave, §16, the Same t ru e for 
living forces of all the particles on any spherical sur- an r spherical 
face whose centre is the point of primitive disturbance 
must be equal to each other ; for the velocities are equal, 
and the medium being of homogeneous density, the masses 
of the particles have the same measure. 

Denote by R the radius of any spherical surface in-illustration; 
termediate between the interior and exterior limits of 
the wave in any assumed position, by n the number of 
particles on the unit of surface, then will the number 
of particles on the entire sphere be 






56 


NATURAL PHILOSOPHY. 


Number of 
particles on a 
spherical surface; 


Sura of their 
living forces; 


Same for another 
spherical surface; 


Living forces 
equal; 


Consequence; 


Role first; 


Rule second. 


n . 4 n R?, 
and the sum of their living forces 

n . 4 tt R 2 . m V 2 ; 

in which V denotes the velocity common to all the par¬ 
ticles, and m the mass of a single particle. 

For another spherical surface, whose radius is R', and 
the common velocity of whose particles is V', we will 
have 

n . 4 7r R ’ 2 . m ' V' 2 . 

Now, if these spherical surfaces occupy the same rela¬ 
tive places in the wave in any two of its positions, be 
their distances from the centre of disturbance ever so 
different, these living forces must, from what is said 
above, be equal; whence we have, after dividing out 
the common factors, 

R 2 . m V 2 = R ' 2 . m V ' 2 


or resolving into a proportion 


mV 2 : m V ' 2 


1 . 1 __ 
R 2 * R 2 


That is to say, the intensity of sound varies inversely 
as the square of the distance to which it is transmitted . 

Again, the particles describe their orbits in equal 
times; their greatest velocities will, therefore, § 16, be 
proportional to their greatest displacements, and the in¬ 
tensity of sound to the squares of these same displace¬ 
ments. 


§ 54. The greatest distance to which sounds are audi¬ 
ble does not admit of precise measurement. It depends 



ELEMENTS OF ACOUSTICS. 


57 


principally upon the absolute intensity of the sound itself, 
the nature of the conducting medium, and the delicacy media; 
of hearing possessed by individuals. Generally speak¬ 
ing, a sound will be heard further, the greater its ori¬ 
ginal intensity, and the denser the medium in which it 
is propagated. 

The greatest known distance which sound has been Greatest known 
carried through the atmosphere is 345 miles, as it is distance in air; 
asserted that the very violent explosions of the volcano 
at St. Yincent’s have been heard at Demerara. Sound an(1 Iouder on 
travels further and more loudly in the earth’s surface earth ’ s 6urface 
than through the air. Thus, for instance, in 1806, the 
cannonading at the battle of Jena was heard in the open 
fields near Dresden, a distance of 92 miles, though but 
feebly, while in the casements of the fortifications it was 
heard with great distinctness. So also it is said that 
the cannonading of the citadel of Antwerp, in 1832, was Instances; 
heard in the mines of Saxony, which are about 370 
miles distant. 

When the air is calm and dry, the report of a musket Ee P° rt of a 
is audible at 8000 paces; the marching of a company 
may be heard on a still night, at from 580-830 paces 
off; a squadron of cavalry at foot pace, 750 paces ; trot- Marchofcavalr yJ 
ting or galloping at 1080 paces distant; heavy artillery, or artillery, 
travelling at a foot pace, is audible at a distance of 660 
paces, if at a trot or gallop, at 1000 paces. A power¬ 
ful human voice in the open air, at an ordinary tem¬ 
perature, is audible, at a distance of 230 paces, and Human voi< *- 
Captain Parry tells us that in the polar regions a con¬ 
versation may be easily carried on between two persons 
a mile (?) apart. 



58 


NATURAL PHILOSOPHY. 


To find the 
distance of a 
particle from its 
place of rest at 
any instant; 


Supposed 
displacement of 
an assumed 
particle; 


Consequent 
displacement of 
another; 


Displacement 
a function of the 
distance V.t-x. 


MOLECULAR DISPLACEMENT. 

§ 55. Let us now seek an expression for the distance 
of any molecule from its place of rest, at any time, dur¬ 
ing the transmission of wave motion. This displacement 
obviously depends upon the intensity of the disturbing 
cause, the distance of the molecule under consideration 
from the place of primitive disturbance, the velocity of 
wave propagation, and the time elapsed since the primi¬ 
tive disturbance was made. 

Disregarding, for the present, the diminution of the 
amplitude of vibration due to the loss of living force 
in the successive molecules as we proceed outward 
from the source of sound, let 

A , be the point of primitive 
disturbance, and B , the place 
of rest of any assumed mole¬ 
cule. Denote by t a?, the dis¬ 
tance of B from A, and by V 
the velocity of wave propagation. At the expiration 
of the time t , after the instant of primitive disturbance 
at A , let the wave front be at W] and the molecule at 

B , be disturbed by the distance B b. The distance of W 
from A , will be V. t. 

Now, from the nature of the motion transmitted, any 
other molecule whose place of rest is C, beyond A, must 
experience an equal displacement C c, at the expiration 
of the time t -f t\ which is as much in excess over the 
time required for the wave front to reach <7, as the 
time t , was over that required to reach B. In other 
woids, the displacements must be equal for successive 
molecules whose places of rest are at equal distances 
behind the wave front • and hence the displacement 
must be a function of this distance, that is, of V.t - x\ 
and denoting the displacement by d , we may write 


Fig. 21. 





ELEMENTS OF ACOUSTICS. 


59 


d ~ F ( T 7 ". t — X) ; First value of 

displacement 

in which F\ denotes the form of tlie function to be em¬ 
ployed. 

Moreover, from the definition of a wave, the nature Function 
of the function F\ must be periodic; that is to say, it penodic ’ 
must, within a given interval of time, pass through all 
its possible values, and resume and repeat these values 
in the same order during the following equal interval 
of time. This is a property possessed by the circular 
functions, and hence we may write the above 


y sm 




Its form ; 


in which 2 7r, denotes the circumference of a circle whose 
radius is unity, and y the radius of the small circle of 
which the sine of some one of its arcs will give the dis¬ 
placement sought. The radius y, is equal to the greatest 
displacement of a molecule in the same wave; for, a simple 
inspection will show that the function takes its maxi¬ 
mum value when the quotient, 

When a 
maximum; 

becomes equal to one-fourth, or to any odd multiple of 
one-fourth; the value being in that case 

y . sin 90°, or y sin 270°, or y . sin 450, &c. = y. Maximum vaiu& 

But the intensity of sound diminishes as the square of Faw of variation 
the distance from its source increases; and the intensity g^ en81ty ° f 
being directly proportional to the square of the greatest 
displacement, §53, if a, denote the radius of the small 
circle at the distance unity from the source of primitive 
disturbance, we have 


V.t-x 

—i— > 





60 


NATURAL PHILOSOPHY. 


Expression of 
the law; 


Radius of the 
circle whose sines 
give the 
displacements; 


Second value of 
displacement; 


When it is zero; 

When a 
maximum. 


Arbitrary 
quantity; 


Final value of 
displacement; 


1 . 

(l ) 3 * a ? 2 


a 2 : y 2 


a 2 

x * 


or 



which substituted for y above, gives 


d = 


^..sin |2 k . 

x L 


3F] 


The quantity V. t, denotes the linear distance of the front 
of the wave or pulse from the source; V.t — x, the dis¬ 
tance of the molecule’s place of rest from the wave front; 
and when this distance contains the lengths, either a whole 
number of times, or a whole number of times plus one- 
half, d becomes zero. When the remainder, after the divi¬ 
sion of V.t- a?, by A, becomes \ or £, &c., the value of d 
a 

becomes —its maximum value. 
x 

If the arc 


be increased by an arbitrary quantity A, it is plain that 
we may assign to A, such a value as to cause any given 
displacement, and therefore the maximum displacement, to 
occur at a given place and time. Introducing this arbi¬ 
trary quantity, we finally have the general equation 


a T V.t— x 1 

d== x ' 8ln + A \ • • • (19) 


in which —, determines the intensity of the sound; A, its 
x 






ELEMENTS OF ACOUSTICS. 


61 


pitch; and A , the particle whose place of rest is at a dis- Meaning of the 
tance x from the source, to have any particular displace- quantltl0S - 
ment at the expiration of the time t. 


INTERFERENCE OF SOUND. 

§ 56. We have seen, in Mechanics, that a body may be Body animated 
animated by two or more motions at the same time; that by t two or more 
the ultimate result of these motions, as regards the body’s 
position, will be the same as if these motions had taken 
place successively; and that one or more of these motions 
may be destroyed at any instant without affecting in any 
wise the others. These coexistent motions, estimated in any 
given direction, become, as it were, superposed upon each Coexistence an(1 
other, and when very small, give rise to a principle known superposition of 

. 1 • . -j •, • n 77 . • ft small motions! 

as the “ coexistence and superposition oj small motions ; a 
principle most fruitful of results in sound and light. By 
it we are taught that when the excursions of the parts of whatit teaches; 
a system from their places of rest are very small, any or 
all the motions of which, from any cause, they are suscep¬ 
tible, may go on simultaneously without disturbing one 
another. 

The truth of this important principle will appear from Its trnth 
its application to the particular case in question. 

It has been shown, §4, that when a molecule of any 
body is very slightly disturbed from its place of rest, 
as in the case of sound, the forces exerted upon it by the 
surrounding molecules give rise to a resultant whose in¬ 
tensity is proportional to the amount of displacement. 

This displacement may arise from the action of a sin¬ 
gle or from several causes operating at the same time; 
but in every case, the expression which gives the value 
of the resultant action must be a function of those which 
express the values of the partial actions, and, like each Explanation, 
of these latter functions, being proportional to the dis¬ 
placement it is capable of producing must, as well as 



62 


Partial, as well as 
entire functions 
must be linear; 


Conclusion; 


Illustration; 


Partial waves; 


Construction of 
resultant wave. 


NATURAL PHILOSOPHY. 


the partial functions, be linear. In any such function, 
if we attribute a slight change to one of the disturbing 
causes, the corresponding change in the displacement 
must be proportional thereto; and whether the change 
in all the partial causes, or in the functions which 
measure them, be simultaneous or successive, the final 
result will be the same; for, the change in the entire 
function in the first case must be equal to the algebraic 
sum of the partial changes in the second. To those fa¬ 
miliar with the calculus, it will be sufficient to say, that 
the first power of the total differential of the sum of a 
number of functions, is always equal to the first power 
of the sum of the partial differentials. 

We conclude, therefore, that the function which gives 
the displacement may be broken up, so to speak, into 
several partial functions equal in number to that of the 
disturbing causes; that these partial functions will be 
similar to each other and to the entire function; and 
that this latter will be equal to the algebraic sum of the 
former. (Analytical Mechanics, §§ 204 and 306.) 

§ 57. To illustrate: 
let the straight line 
A B , be the locus of a 
series of molecules in 
their positions of rest; 
the fine waved line 
ab , that of the same 
molecules at a particu¬ 
lar instant of time, when disturbed and thrown into a 
wave by the action of some single cause ; and the waved 
line a' V, that of the same molecules at the same in¬ 
stant had they been thrown into a different wave under 
the operation of some other insulated action. If these 
disturbing causes had acted simultaneously, the locus of 
the disturbed molecules would be represented by the 
heavy waved line X I 7 ", constructed in this wise: At 
the various points of the line A B , erect perpendiculars 


Fig. 22. 





ELEMENTS OF ACOUSTICS. 


G3 


and produce them indefinitely ; lay off from A B, on Resultant curve; 
these perpendiculars, distances equal to the sum or dif¬ 
ference of the corresponding ordinates of the component 
curves, according as these curves intersect the perpen¬ 
diculars on the same or on opposite sides of the line 
A B ,—the points thus determined will he points of the 
resultant curve, which will give the law of displacement 
at the instant of time in question. Were there three, same for three or 
four, &c., component curves, the resultant curve would components 
be determined by the same rule. 


§ 58. Taking; it, then, as a fact, that the disturbance „ , 
of every molecule produced by the coexistence of two of two equal 
or more causes will be the algebraic sum of the dis- ' vaveson a 

° particle; 

turbances which they would produce separately, let us 
consider the nature of the displacement produced by 
the superposition of the action of two waves of the 
same length on the same molecule, the waves being 
supposed to come from any directions whatever. 

We shall have for the displacement of the molecule 
by the first wave, Eq. (19), 


a! • r 0 V. t—x i 

d = — . sm 2 n. -_i_ a'\ . 

x L X J ’ 

and by the second, 


(20) Displacement by 
the first wave; 


d" = .smhn.LL^+A"]; 

x L X J 


( 21 ) 


Same by the 
second; 


in which a! and a”, 
determine the intensi¬ 
ties of the sound in 
the two waves at the 
unit’s distance ; and A’ 
and A!\ the places of 
the maximum displace¬ 
ment at the expiration 
of the time t. 



Eustiatlon; 







64 


NATURAL PHILOSOPHY. 


Operations 

indicated; 


Sum of 
displacements; 


Supposition; 


Notation; 


Total 

displacement; 


Same; 


Transformations 


Reductions; 


Conclusions; 


Taking the sum, and developing the circular function 
by the usual formula for the sine of the sum of two 
arcs, we find, after reduction, 

*4*/sin [2 ». .cos^.Zp]. 

and making, 

a . cos A = a' cos A' + a” cos A!\ . . . (a) 
a . sin A = a' sin A! -f a " sin A", ...(b) 

the above becomes, after writing d for the total displace¬ 
ment, 

d — j^cos A . sin ^2 tt + sin A . cos ^2 n ; 

replacing the quantity within the brackets by its equal, 
viz.: the sine of the sum of the two arcs, we have 

d = +A] . . . (22) 

X L A J 

Squaring Equations (a) and (b) and taking the sum, we 
find, 


a 2 = a' 2 4- a," 2 + 2 a' a" cos (A'— A") . . . (23) 


and dividing Equation (i b ), by Equation (&), we obtain 


tan A 


a ’. sin A! + a", sin A" 
a'. cos A! + a", cos A" * 


• • ( 24 )- 


From Equation (22) we see that the length of the 
resulting wave is the same as that of the partial waves; 
but the value of A in that equation differing from A\ 












ELEMENTS OF ACOUSTICS. 


65 


and A!\ Equation (24), shows that the maximum dis- Timeof 
placement for a given molecule does not take place displacement in 
with the same value of t , as for either of the compo- resultantwaTe; 
nent waves. 

The maximum displacement which determines the 

X 

intensity of the sound, in the resultant wave, is given by 
Equation (23) to he 


a 

x 


1 _ 

x 



+ a"* +2a'a".cos(A' - A") 


(25) General value of 
this 


which depends upon the arc 
A! — A". Its greatest va¬ 
lue is obtained by making 
A' — A" = 0, in which case 
we have 


displacement; 



a a! 4- a ". 

— = -• Greatest value; 

X X 


its least value results from 
making A! — A" = 180°, in 
which case 


Fig. 24. 



When this value 
is least; 


a __ a 
x 


x 


Least value; 


In the first case Equation (24) gives 

tan A = + a ) v 8 ! 11 _ tan A' = tan A "; 

(a' + a"). cos A' # 


First case; 


whence A , is equal to A\ and to A", and the maximum conclusion- 
displacement will occur at the same place and at the 
same time in the resultant wave, and in both compo¬ 
nent waves. 

In the second case, if we substitute in Equation (24) 

A' = 180° + A”, we find 

5 









66 


NATURAL PHILOSOPHY. 


Second case; 


Conclusion; 


Intensity of 
sound supposed 
equal in 
component 
waves; 


Consequence; 


Reduction; 


Value of 
arbitrary 
constant; 


Supposition; 


Illustration. 


tan A 


(a"—a'), sin A” 
(a"—a') cos^4" 


=tan ^"=tan(^'—180°)=tan.4'; 


that is, A is equal to one at least of the arcs A’ and 
A'\ and the .greatest displacement in the resultant wave 
will occur at the same place and time as in one of the 
component waves. 


§ 59. If the intensity of 
sound in the component 
waves be supposed equal at 
the place of superposition, 
then will a' — a!\ and Eq. 
(25) becomes 


Fig. 25. 



a 

x 


2 a' A' - A” 

-cos_±_ 


x 2 

and Equation (24) reduces to 

tan A = sin 4L + sin A " 
cos A' + cos A" 

A' + A” 


tan 


A'+ A’ 
2 


or. 


A = 


2 


JB 


. t/JB) 




When A' — A" — 0 , then will Eq. (26) give 


- = —, and A = A’ = A "; 

X X 


that is, the intensity of sound 
in the resultant wave is quad- 
rujile that in either of the 
equal component waves; and 
the greatest displacement 
will occur at the same time 
and place in the component 
and resultant waves. 


Fig. 26. 
















ELEMENTS OF ACOUSTICS. 


07 


If a! and a" continue equal, and we make A'—A"= 180°, supposition; 
tlien will Equation (26), give 



(26)'. 


or in words, one of tlie equal 
sounds will destroy the other. 

Thus it appears that two 
equal sounds reaching the 
same point may he in such 
relative condition that one 
will wholly neutralize the other, and the two produce 
perfect silence. This phenomenon is called the Inter - Interference of 
ference of sound. 

With any other values for A! and A" than those which 
give A ' — A" = 180 or 0°, Eq. (26), shows that 


Silence 

Fig. 27. produced; 



a 2 a\ 

— \-j 

x x 


Result of partial 
coincidence of 
two sound waveflk 


that is, the sound in the resultant wave is less than quad¬ 
ruple that in either of the equal component waves. 


§ 60. To ascertain the precise relation between two 0011(33110119 that 

u ...... . will cause two 

equal waves, which will cause one to destroy the other, eq uai waves to 
make, in Equation (20), neutralize each 


A' A” ± 180° = A" ± * 


and we have 


<V 


a! r 

= —- sin 2 
x L 


V.t—x 


+ A" ± 


•] 


but 


2 . X Transformations; 

2. A 







68 


NATURAL PHILOSOPHY. 


Resultant 
displacement; 


Conditions for 
interference. 


When waves 
interfere only at 
point of union. 


Same 

considerations 
applicable to 
three or more 
equal waves; 


Equations to be 
used; 


and this substituted above, the equation becomes 


d'= - .sin fo«. 
x L 


V. t-x ± 
X 



which becomes identical with 
Equation (21) by writing 
x , for x qp \ X. That is to 
say, one wave will destroy 
another of equal length and 
intensity, if, starting from 
the same origin, in the same phase, they meet, after trav¬ 
elling over routes that differ in distance by half a wave¬ 
length. 

And since a difference of route equal to any whole num¬ 
ber of wave lengths produces no difference of phase in the 
undulation, it is obvious that a difference of route equal to 
any odd multiple of half a wave length, produces the same 
effect as a difference of a single half. 

Thus, two waves will destroy one another, if they be 
of the same length, have the same maximum molecular 
displacement, travel along the same route, and have, at 
any point, opposite phases. If they travel over different 
routes and meet, they can only interfere at the point of 
union. This mutual destruction of two waves, having op¬ 
posite phases at their place of union, is illustrated at § 52. 

§ 61. The same process of combination may be ap¬ 
plied to three, four, &c., waves of equal lengths. Thus 
let there be the Equations 

d' = £' Bin [2* 

XL X J 

• ■'"=?»» [ a ' 


Fig. 27. 








ELEMENTS OF ACOUSTICS. 


69 


^"' = ~ sin [2 *.L±? + A m ] 


Operations 

performed; 


adding these, developing the 
sine of the sum of the two 
arcs within the brackets, col¬ 
lecting the common factors 
and denoting the resultant 
displacement by d , we have 



g Illustration; 


id a • n Y.t-X a • a F. t-X 

d= — cos A . sin2. - 1 -smA. cos 2_, 

X X X 


X 


Eesultant 

displacement; 


or 


d= ®.sin [ 2 *. VA ~ X + A\ ; 
x L k J 


The same; 


in which 


a cos A = cos A/ -j- a" cos A" + a m cos A!” — X 
a sin A = a' sin A! + a" sin A" + a'" sin A" f = Y 


a = 


Y 


■■y/2P+Y*; tan A = • 


Notation, 


§ 62. Although it is possible for two waves of sound, 
whose lengths are the same, to neutralize each other, it is 
not so when the 
waves have un¬ 
equal lengths; 
for, Eq. (22) 
was deduced 
by making Y 
and X the same in the two component waves, the sum 
of d r and d" being in that case reducible. If these con¬ 
ditions were not fulfilled, this sum would not be reducible, 
and there would be the two arcs 



Two unequal 
waves cannot 
neutralize each 
other; 


Illustration; 











70 


Natural philosophy. 


Explanation; 


Conclusion 
respecting 
unequal waves; 


Any disturbed 
particle causes 
subsequent 
disturbance in 
another; 


Same true for all 
particles in a 
wave front; 


Illustration; 


2^ and 2 *——, 


in the final value for d, with different coefficients, which 
could not be made equal to zero at the same time. The 
values of F, will, to be sure, be the same in any two 
waves of sound, but this need not be so with those of X ; 
and in waves which produce light, in which subject we 
shall have most occasion to refer to the doctrine of in¬ 
terference, the values of F, as well as those of X, may 
differ. The discussions of waves of different lengths may, 
therefore, be kept perfectly separate, as the combined 
effect of such waves will be the same as the sum of 
their separate effects, without the possibility of their 
destroying or modifying one another. 


NEW DIVERGENCE AND INFLEXION OF SOUND. 


Fig. 20. 


§ 63. We have seen that every disturbance of a mole¬ 
cule at one time is truly a cause of disturbance of an¬ 
other molecule at some subsequent time. All the mole¬ 
cules in a wave front become, therefore, simultaneously 
centres of disturbance, from each one of which a wave 
proceeds in a spherical front, as from an original dis¬ 
turbance of a single molecule. Thus, 
in the wave front A B, a molecule 
at x becomes a new centre of dis¬ 
turbance as soon as the wave front 
reaches it; and if "with a radius 
equal to V.t a circle be described, 
this circle will represent a section c 
of the spherical wave front proceed¬ 
ing from a?, with the velocity V , at 
the end of the interval of time de¬ 
noted by t. And the same being 
true tor the molecules a?', a/', &c., of the primitive 
wave, there will result a series of intersecting circles 






ELEMENTS OF ACOUSTICS. 


n 


having equal radii, and the larger circle A! B '. Construction of 
tangent to all these smaller circles, will obviously be a front; 
section of the main wave front at the expiration of the 
interval t, after it was at A B. Any molecule situated 
at the intersection of the smaller circles will obviously 
be agitated by the waves transmitted to it from mole- Kesultant 
cules at their respective centres; and the resultant dis- displacement of 
placement will, §55, §56, be the algebraic sum of the a partlcle ' 
displacements due to each when superposed. 

Hence, to find the disturbing effect of any wave upon 
a given molecule at a given time, divide the wave into 
a number of small parts, consider each part as a centre M e ‘ 
of disturbance , and find by summation the aggregate of 
all the disturbances of the given molecule by the waves 
coming from all the points of the great wave. 

The cause which makes the disturbance of a single 
molecule at one instant the occasion of the simultaneous 
disturbance of an indefinite number of surrounding mole- „, , , 

° Principle of new 

cules at a subsequent instant, is called the principle of divergence 
new divergence , of which frequent use will be made in 8tated; 
the subject of light. 


§ 64. Let us trace the consequences of this principle Its application to 
in its application to the passage of sound through aper- 80 und thTougi. 

apertures and 
around corners; 


Fig. 81. 


tures and around the edges 
of objects. Take a parti¬ 
tion M N, through which 
there is an opening A B , 
and suppose a spherical 
wave of sound to proceed 
from a centre C. Only that 
portion of the wave which 
comes against the opening 
can pass through, and the wave front on the opposite 
side of the partition will be found by taking the diffe- Illastratioa; 
rent points of the segment A B , within the opening as 
centres, and radii equal to V. t, and describing a series 
of elementary arcs, and drawing a curve tangent to them 





72 


NATURAL PHILOSOPHY. 


Explanation an<3 
construction; 


Fig. 81. 



Sonnd that is not 
reinforced by 
particles from the 
primitive wave; 


Sectors wherein 
the sound is due 
to superposition 
of waves from the 
edges; 


all. That portion of this tangent curve included be¬ 
tween the lines C A' and 
GB\ drawn from <7, and 
tangent to the limits of the 
opening, will obviously be 
the arc of a circle having C 
for its centre. The elemen¬ 
tary circles described about 
the limits A and B as cen¬ 
tres, cannot be intersected 
at points exterior to the angle A' CB' by those described 
with equal radii from points of the wave front lying 
between A and B; the wave front within the angles 
A' A M and B’ B N, will have their centres at A and 
B respectively; and the' sound proceeding from these 
points will be diffused over the arcs A' M and B r N 
without reinforcement from molecules of the same primi¬ 
tive wave. 

But other waves from C reaching the opening in suc¬ 
cession, a spherical wave diverging from B , and of which 
the radius is B 0 , will be overtaken by a subsequent one 
from A, having for its radius AO/ so that, the intensity 
of sound in the angle A' A M will result from the super¬ 
position of the disturbances from B and A. The same 
will be true of the sector B' B 2T. 


Intensity 
increased by 
coincidence; 


Decreased by 
interference. 


Now, if B 0 — A 0, be equal to X, 2 X, 3X,... n\ in 
which n is a whole number, then will the intensity of the 
sound be increased above that due to either of the com¬ 
ponent waves. But if B 0— A 0, be equal to IX, |X 
.... (n + J) X, n being still a whole number, the compo¬ 
nent waves will interfere at (7, and the intensity of the 
sound will be lessened at that point by the prevention 
there of the disturbance due to either of these two 
component w r aves. 


Points taken Taking another molecule B „ nearer to A , the wave 
between the from Z?, will interfere with the wave from A , but at 

edges 5 . , ~ _ . 7 

a point Op nearer to the partition, m order to pre¬ 
serve the difference B t 0 J - A the same as before, 




ELEMENTS OF ACOUSTICS. 


73 


to wit, ( n + |) X. Assuming other points in succession construction; 
nearer to A , we shall find the interference to take 
place at molecules still nearer to the partition; and 
finally, when we come to a molecule Q , in the main 
wave front whose distance A Q, from A is equal to 
(n + i) X, the interference will occur at a molecule situ¬ 
ated against the partition at P. 

Now, making n = 0, in which case A Q will equal 
\ X, and applying 

\ X from a to a t , Flg * 82 ‘ 

the waves from a ^ I1Iustrat5on; 

and will inter¬ 
fere at P. Applying £ X, from b to b J , the waves from 
b and b J will also interfere at the partition; and in the 
same way it may be shown that all the partial waves 
from molecules in the distance A Q, will interfere with 
those from the molecules in the distance Q P, QD 
being equal to A Q. Commencing the same process at P, 
we see that the opening may be such that on applying 
\ X from a' to a! t , this latter point a\ may be in the posi- Explanation of 
tion from which there can be no new divergence to inter- results; 
fere with that from a '; and the same for the whole of 
the arc D P, of the main wave. This latter is, therefore, 
left, as it were, undisturbed, and sound from it may or 
may not be audible at jP, depending upon the extent of 
this arc and the intensity with which the sound reaches 
the opening. 

The distance A Q is equal to £ X. But X, we sound heard at 
have seen, § 48, varies with the pitch, whence the sound P art ^ ion de P ends 

707 r 7 on pjtch, and size 

heard at P, will depend upon its pitch and the size of evening, 
of opening through which it may pass. 


§ 65. From what precedes we see that at the line Acoustic shado-vr 
A 0 , Fig. 31, there begins, as it were, an acoustic shadow , cast 
which deepens more and more as we approach the 
partition towards P, where the sound becomes least Inflexion of 
audible. This bending of sound around the edges of an 
opening is called the Inflexion of sound. 





74 


NATURAL PHILOSOPHY. 


Case of sound 
bending around 
corners; 


Explanation; 


No perfect 
neutralization; 


Grave sounds 
more audible 
than the acute; 


Case of little 
inflexion. 


§ 66.When the opening is continued indefinitely in one di¬ 
rection only, we have the case of sound bending around a cor¬ 
ner. But when the open¬ 
ing is continued indefinite- Fig - 3S - 

ly in one direction, there • ^ mim i nii i imiii i iimiimm ur ^—q j? a u>ji* 

can be no arc of the main 

wave as D B , (last figure), without a corresponding arc 
T> i B t , further on, to neutralize it in part at least by 
interference, and hence, were the component sounds of 
the same intensity at the point of superposition, they 
would produce perfect silence, and no sound could be 
heard at P. 

The sound from the main wave is of the same intensity 
throughout on reaching the corner; the new diverging 
waves leave their respective centres, which are distri¬ 
buted along the front of the main wave, with equal 
intensities; they can only interfere after having travel¬ 
led over routes which differ by |X; the intensity of 
sound varies inversely as the square of its travelled dis¬ 
tance ; and the intensities cannot be equal at the places 
of interference, and therefore can only partially neutralize 
each others’ effects. This is shown by Equation (26)', in 

which -- is zero, only because a?, under the conditions 

X 

there imposed, is the same denominator for a ' and a". 
In sound, X varies, as we have seen, § 48, from a few 
inches to many feet, and as the difference of intensities in 
the interfering waves will be greater as X is greater, the 
graver sounds would be heard, under the circumstances 
we have been considering, more audibly than the more 
acute. If the lengths X, were insensible in comparison 
with the route travelled, there would be but little in¬ 
flexion ; since, in that case, the intensities of sound in the 
interfering waves would be sensibly the same, and it 
would require but a slight obliquity from the direct 
course of the main wave to make a difference of route 
B 0 — A 0, Fig. 31, equal to |X, necessary for one wave 
sensibly to destroy the other. 




ELEMENTS OF ACOUSTICS. 


75 


An auditor placed behind a wall 
at i 3 , would hear the bass notes 
from a band of music playing at a 
position A on the opposite side, much 
more distinctly than the acute notes. 

At jP, the notes of the tuba, for 
instance, might be heard distinctly, 
while those of the octave flute would 
be lost to him. In passing from the 
position P to 0 , he would catch in 
succession the higher notes in order of the ascending 
scale, and finally, when he attained a position near the p osition whenc . 
direct line A 0 , drawn from A, tangent to the corner, he a11 the 
would hear all the instruments with equal distinctness, if Iqulnyludiw™ 
played with equal intensity and emphasis. The facts 
and explanations here given have an important applica¬ 
tion in the subject of optics. 

If we suppose the lengths of sonorous waves propagated 
through water to be much shorter than those through the 
air, we have here a full and satisfactory explanation of 
the phenomenon observed by M. Colladon, mentioned at 
the close of § 39. Indeed, taking the acoustic shadow Foregoing 
there referred to as established, it must follow as a conse- ded " ctlons 
quence, from the principle of new divergence, that the experiment* 
lengths of the sonorous waves in liquids are shorter than 
in air.* (Analytical Mechanics, § 348.) 



REFLEXION AND REFRACTION OF SOUND—ECHOS. 

§ 67. There is no body in nature absolutely hard and Disturbed 
inelastic. Whenever, therefore, the molecules of a vi- particles of one 
bratinsr medium come within the neutral limits of those bod . va ^ tate 

» ... those of anotbe 

forming the surface of any solid or fluid, they will and transmit a 
agitate the latter with motions similar to their own, and pulse * 

<D ' 

a pulse will be transmitted into the solid or fluid with 
a velocity determined by its density and elastic force. 

*See Appendix No. 1. 





76 


NATURAL PHILOSOPHY. 


References; 


Velocity of a 
particle; 


Excess of 
condensation; 


Expressed by an 
equation; 


Velocity of a 
particle; 


Rule for 

homogeneous 

media. 


When a particle 
will come to rest 


§ 68. Referring to the transmission of sound through 
air, and resuming Equation (2)', we have, after substituting 
the value of V, as given by Equation (3), 




x — x t 
x 


Row, by reference to §34, it will be seen that 


x — x / 
-? 

x 


expresses the excess of condensation on one side of a 
molecule over that on the opposite side. Making 


the above Equation may be written 

V =C-\Z^- .( 28 ). 


In the same homogeneous medium E and D are con¬ 
stant, whence we conclude that the actual velocity of a 
molecule , which is the same as that of the stratum to 
which it belongs, is directly proportional to the excess 
of condensation on one side of it , over that on the oppo¬ 
site side. 

When, therefore, by the forward movement of a mole¬ 
cule the condensation becomes equal on opposite sides, 
the molecule comes to rest, and remains so till again 
disturbed by some extraneous force. This explains why 
it is that a pulse transmitted through a medium of uni- 








ELEMENTS OF ACOUSTICS. 


77 


form density sends back no disturbance, but leaves every Living force 
molecule behind in a state of rest. The living force im- transferred - 
pressed upon any given stratum is transferred to the next 
one in front, and this to the next in order, and so on in¬ 
definitely. 

i 

§ 69. When the 
stratum A B is mo¬ 
ved by some source 
of disturbance to 
A' B\ the stratum 
CD will move in 


c" 


ir 


Fig. 35. 

b b' _ a 


Stratum 
disturbed; 


A' K’ 


ir 




A A' 11 


zr 


A pulse 
transmitted in 

the same direction, and a pulse will be transmitted on- direction of 

disturbance; 

ward towards W, the excess of condensation being on the 
same side of the moving stratum as the place of the ori¬ 
ginal disturbance. But a shifting of the stratum A B 
to the position A! B\ leaves the excess of condensation And also one in 
which acts on the stratum CD’ on the opposite side direction • 
from A B ; the stratum O' D’ will therefore close in 
upon A’ B\ and the same occurring in succession with 
all the strata on the side towards W\ a pulse will be trans¬ 
mitted in an opposite direction from that which begins 
with the motion of C D. Thus, every case of an original 
disturbance of a molecule will give rise to two pulses pro- Every 
ceeding in opposite directions, with the same velocity, the ^^^pubes; 
two pulses differing only in this, viz.: in the one the 
wave velocity will be in the same direction as that of Theirdifference - 
the molecules, and in the other in an opposite direc¬ 
tion. 


§ 70. The elastic force E, of two media in contact and Elastic force of 
at rest, must be the same; otherwise motion would ensue. contact and at 
When, therefore, in the progress of a pulse, it reaches a rest 
stratum X T, of a density or elasticity different from that 
of those which precede it, Equation (28), shows that for the 
same excess (7, of condensation, the velocity of the stratum Effect when the 
will be altered; that is, the actual motion of the molecules “^tslneT 111 
will either be accelerated or retarded. If the new stra- greater density. 










78 


NATURAL PHILOSOPHY. 


Effect when 
moving stratum 
meets one of less 
density. 


turn be of increased density, the next preceding stratum 
K Z, will be checked in its progress by the greater mass 
of X I 7 ", and brought to rest before it reaches its neutral 
distance from that behind; the excess of elastic force 
thus retained will react upon the next preceding stratum 
which has already come to rest, and will thus give rise 
to a return pulse in which the velocity of propagation 
and that of the molecules will be in the same direction. 
If, on the contrary, the new stratum have a dimin¬ 
ished density, the motion of KL will be accelerated, the 
density in front of the next preceding stratum will be¬ 
come less than that between those behind which have 
come to rest; these latter strata will therefore move for¬ 
ward in succession, and thus a return pulse will be pro¬ 
duced as before, but .with the difference, that the velocity 
of propagation and that of the molecules will be in oppo¬ 
site directions. 


Wave meeting a 
medium of 
different density 
is resolved into 
two; 


Cause of this 
resolution. 


Incident, 
refracted, and 
reflected pulses. 


§ 71. It follows, therefore, that when a pulse or wave 
of sound in any medium reaches another medium of 
greater or less density, it is at once resolved into two, 
one of which proceeds on through the second, while the 
other is driven back through the first medium. 

This division of an original pulse into two others, arises 
entirely from the reciprocal action of the two media on 
each other. If the media be perfectly elastic, there can 
be no loss of living force, and the sum of the intensities of 
sound in the component pulses will be equal to that of the 
original pulse. If the media be not perfectly elastic, 
there will be a loss of living force, and the sum of the 
intensities of the component pulses will be less than that 
of the original pulse. 

The original pulse is called the incident; that transmit¬ 
ted into the second medium, the refracted; and that 
driven back through the original medium, the reflected 
pulse. 

To an ear properly situated, the reflected pulse will be 
audible, and is, for this reason, called an echo. The 6ur- 


Echa 



ELEMENTS OF ACOUSTICS. 


79 


face at which the original pulse is resolved into its two Aviating 
component pulses, is called the deviating surf ace. 

§ 72. To find the law which regulates the direction of Direction of the 
the reflected pulse; let A M be a 
portion of the front of an incident 
spherical pulse, so small that it may 
be regarded as a plane. Draw M A", 

A! N and A 0 , normal to the pulse, 
and suppose the latter, moving 
in the direction from N to A\ to 
meet tfie face E G of a second me¬ 
dium. Each molecule of the pulse 
as it recoils from the surface E G, 
becomes the centre of a diverging 
spherical pulse which will, Eq. (28), 
be propagated with the velocity of 
the incident pulse. Accordingly, when the portion ATExplanation and 
reaches the face of the second medium at A", the por- constructlon; 
tion A will have diverged into a spherical pulse whose 
radius is AB — A" M. In like manner, if A' M' be 
drawn parallel to A M j the portion diverging from A! 
will, in the same time, have reached the spherical pulse 
whose centre is A! and radius A! B' = A"M’. The same 
construction being made for all the points of the incident 
pulse as they come in succession to the deviating surface, 
the surface which touches at the same time all these 
spherical surfaces will obviously be the front of the re¬ 
flected pulse. But because A' B' and A B are respec¬ 
tively proportional to A! N and A" J/, and as this is true 
for any other similar lines drawn from points of the 
deviating surface to the corresponding points of the in¬ 
cident and reflected pulses, this tangent surface is a plane. Incident ana 
Moreover, since A B is equal to If A", and the angles reflected p u1s68 
A M A" and A" B A are right, the angles MA A" and ^es with 
BA" A are equal, and the incident and reflected pulses deviating surface, 
make equal angles with the deviating surface. 

Any line which is normal to the front surface of a 





80 


NATURAL PHILOSOPHY. 


Kay of sound 


Angle of 
incidence; 


Angle of 
reflexion; 


These angles 
equal. 


Fig. 87. 


pulse, is called a ray of sound. The angle NAD, 
which the normal to the incident pulse makes with the 
normal to the deviating surface, is called the angle of 
incidence. The angle BAD, which 
the normal to the reflected pulse 
makes with that to the deviating 
surface, is called the angle of re¬ 
flexion / and because the angle 
made by two planes is equal to 
that made by their normals, we 
conclude from the foregoing, that 
in the reflexion of sound, the angles 
of incidence and of reflexion are 
equal. 



Direction of the 
refracted pulse 
determined; 


Construction and 
explanation. 


Fig. sa 


§ 73. The law which determines the course of the re¬ 
fracted pulse is equally simple, t and is deduced in a man¬ 
ner analogous to the preceding. 

Let A M be an inclined element¬ 
ary plane pulse, incident upon a de¬ 
viating surface E G, at any instant. 

In the interval of time during which 
the point M is moving from M to A", 
the agitation which begins at A will 
have reached some spherical surface 
within the second medium of which 
A B is the radius ; and in like man¬ 
ner, the agitation which begins at 
A ', will have reached some spherical 
surface of which A' B’ is the radius, 
by the time the portion of the inci¬ 
dent pulse at M\ will have passed on to A ”; and the 
same of intermediate points of primitive disturbance on 
the deviating surface between A and A ", the first and 
last points of incidence. The surface tangent to all these 
spherical surfaces will be the front of the transmitted or 
refracted pulse ; and because A B and A' B r are respec¬ 
tively proportional to A" M and AN, this surface is a plane. 









ELEMENTS OF ACOUSTICS. 


81 


The angle NAD, made by 
the normal to the refracted 
pulse and that to the deviating 
surface, is called the angle of 
refraction. Denote the angle 
of incidence NAD, which is 
equal to the angle M A A ", 
Fig. 38, by 9 ; and the angle of 
refraction N A D, which is 
equal to the angle A A" B, 
Fig. 38, by 9 '; then will 



• MA" = A" A . sin 9 ; 
AB = A"A.sm 9 '; 


and dividing the first by the second 

MA" _ sin 9 
A B ~~ sin 9 ' ’ 


but A" M and A B , being described in the same time, Explanation; 
the first by the incident, the second by the transmitted 
pulse, are respectively proportional to the velocities in 
the two media. Denoting the velocity of the incident 
pulse by V, and that of the transmitted pulse by V', 
we have 

V _ MA" __ sin 9 
V' ~~ A B ~ sin 9 ' 

whence 

sin 9 = —y sin 9 ', .... (29). 

That is to say, in the refraction of sound the sine of the Euie. 
angle of incidence is equal to the sine of the angle of re¬ 
fraction multiplied into the ratio obtained by dividing 
the velocity before incidence by that after refraction . 


Eatio of 
velocities of 
incident and 
refracted sound 









82 


NATURAL PHILOSOPHY. 


Application to 
air and water; 


Illustration; 


Thus, if sound proceed through the atmosphere at 32° 
Fahr., and be incident upon the surface A B , of water at 
the same temperature, then will V = 1089,42, V'= 
4707,4, and 

Fig. 40. 


v_ _ 

V' ~ 


1089,42 

4707,4 


= 0,23142 


which in Eq. (29) gives 


sin 9 = 0,23142 . sin 9 ' 


or 



sm 9 

p3l42 = sin ^.( 30 > 

Example; Now, suppose the angle of incidence R IN, to be given, 
say 30°. With the point of incidence /, as a centre and 
radius unity, taken from any scale of equal parts, de¬ 
scribe the circumference of a circle; from a table of natu¬ 
ral sines take the sine of 30°, and by means of the same 
scale lay it off from I to II; through H draw II O pa¬ 
rallel to the normal H /, and through the point (9, in 
which this parallel meets the circumference and the point 
of incidence /, draw RI. This gives the incident ray. 
Construction of Divide the sine of 30° by 0,23142, this will give the sine 
incident and 0 f ^ . j a y 0 ff its value from I to II\ and draw IF O’ 

refracted rays. 7 ^ . , 1 

parallel to NI; join the point in which this parallel cuts 
the circumference with the point of incidence 7", and we 
have the direction of the refracted ray 1R'. 

‘ X 

when sound § 74. The sine of an angle can never exceed unity. 
r™cond a8Sint ° ^^ en ) therefore, the angle of incidence becomes so great 
medium; ' that its sine divided by the ratio of the velocities exceeds 
unity, refraction, or which is the same thing, the passage 
of sound from one medium to another in w T hich its velo¬ 
city is greater, becomes impossible. In the case of air 








ELEMENTS OF ACOUSTICS. 


83 


and water, the limit of the greatest angle of incidence Greatest augie of 
corresponding to which we may have any transmission of which60imd can 
audible sound to the second medium, is found from Equa-P assfromairiato 
tion (30) by making sin 9 ' = 1, which gives 


sin 9 = 0,23142 


or. 


9 = 13° 22'. 


Its value. 


§ 75. When the sound is thrown back from the surface Reflected S0U nd; 
separating the two media and continues in the first medium, 
the velocity retains the same value, but its sign will be 
changed. This will make V' = — V, and 


V_ 

V' 


- 1 , 


which reduces Equation (29) to 


sin 9 = — sin 9 ', 


or 


9 = - 9' 

the law of reflexion as given in § 72. 


§ 76. When the pulse proceeds in 
a homogeneous medium from a point 
of disturbance, it takes a spherical 
shape, the normals all meet at the 
centre of the sphere, and the rays 
are then said to diverge from a point, 
in which case the sound becomes 
less intense as it proceeds. 


Ratio of 
velocities of 
incident and 
reflected sound; 

Fig. 41. 


2 - 


Fig. 42. 




Angle of 
incidence equal 
to that of 
reflexion. 


Diverging sound 


Intensity 

diminishes; 






84 


NATURAL PHILOSOPHY. 


Converging 

Bound; 


Intensity 
increases; 


Illustration of 
divergence and 
convergence of 
6©und; 


Decreases in 
loudness before 
reflexion; 


Increases after 
reflexion; 


•Mnvminm 

intensity. 


When in the progress of the pulse it retains its spheri¬ 
cal shape, but any portion of it be¬ 
comes so modified as to present its 
concavity in front, the rays will meet 
at some point in advance, and are 
said to converge / the sound will be¬ 
come louder and louder as it pro¬ 
gresses, and finally, when it reaches 
the point of union of the rays, it will 
attain its maximum intensity ; for in 
this position the living force, which 
was before distributed among the molecules of an ex¬ 
tended pulse, is concentrated in the few molecules of a 
very contracted pulse. 

§ 77. To illustrate, conceive a disturbance to take place 
at the focus F, of an ellipsoid; a 
pulse will proceed from this 
point in all directions. Any two 
rays, as F D and FE, will, from 
the law of reflexion just explain¬ 
ed and the geometrical properties 
of the ellipsoid, pass to the other 
focus F\ as will also the portion 
of the pulse included between 
these rays and which is reflected at the surface D E. 
The living force impressed upon the molecules in the ver¬ 
tex of the angle D F E, will, as the pulse proceeds from 
F, become more and more diffused, and when the pulse, 
reaches the point Z>, this living force will be distributed 
among the molecules of the surface D E r . After re¬ 
flexion, the concavity of the pulse is turned to the front, 
its extent becomes less and less as it approaches the 
second focus, and the living force of its molecules will 
be more and more concentrated, till finally, when the 
pulse reaches the focus F\ the living force of a single 
molecule will be a maximum, and will be capable of pro¬ 
ducing the greatest impression upon the ear. 


Fig. 44. 










ELEMENTS OF ACOUSTICS. 


85 


What has been said of the portion of the pulse within au the sound 
the sector DFE, is equally true of any other sector ^entrateTL 
and of the whole spherical pulse; so that all the sound the other, 
which originated in the focus F, will, after reflexion, be 
concentrated in the focus F'. 

§ 78. When a spherical pulse is incident upon a plane spherical pulse 
deviating surface, it will be easy from the principles now ir J cldent on a 

o i j r r plane surface* 

explained to construct both the refracted and reflected 
pulses For this purpose, let 
A B represent the deviating 
surface; Z>, the point of pri¬ 
mitive disturbance; D C, any 
incident ray; G G and CB, 
the corresponding refracted 
and reflected rays respec¬ 
tively. From the point ./}, 
draw ED , perpendicular to 
the deviating surface. Extend 

the refracted ray C G , back till it meets this line in the 
point H. At the point of incidence C\ draw G M parallel 
to ED. De note the angle of incidence D CM— GDE m ** ti0 *' 
by 9 ; the angle of refraction M G11= G HE by 9 '; the 
distance D E by f ; and the distance HE by/'. Then 
will 

f tan 9 = G E — f tan 9 ' 

whence 



j}/ _y? tan 9 _ /• 

J ’tanV 


sin 9 

cos 9 _ „ sin 9 
sin 9 ' J sin 9 ' 
cos 9 ' 


cos 9 ' 
cos 9 ’ 


Equations; 


making, Equation (29), 



sin 9 
sin 9 ' ’ 


(31). T: 


ransformations • 







86 


NATURAL PHILOSOPHY. 


Operations 
performed; 


Direction of 
refracted ray 
determined. 


Point from 
which the 
reflected rays 
will diverge; 

Deflected pulse 
spherical; 


Diustration; 


and substituting for cos 9 ' and cos 9 , their values 
VI — sin 2 9 ' and VI — sin 2 9 and eliminating sin 9 ' by 

its value , we finally have 
m 


f =f.m. 



(32). 


The distance of the point D from the deviating surface 
and the nature of the two media on the opposite sides of 
the latter being given, the value of f, V and V' will be 
known; and assuming the direction of the incident ray 
D G , the angle 9 also becomes known, and the value of 
f, which determines the point II, will result from 
Equation (32), and the direction of the refracted ray 
HOG , will thence become known. 

For the reflected ray, V and V' become equal with 
contrary signs, and m will be equal to minus unity. This 
will reduce Equation (32) to 

/'= - /; 


that is to say, all the reflected rays will diverge from 
a point D', as far behind the deviating surface as the 
point D of disturbance is in front of it. The reflected 
pulse will, therefore, be spherical. 

From the point D as a centre 
and radius D K\ equal to that 
of the spherical pulse at any 
instant, describe the arc 0II O'; 
this will represent a section of 
the incident pulse by a plane 
normal to the deviating surface. 

Make the distance ED' equal 
to D E\ and with D' as a centre, 
and radius D' K' equal to DII, 


Fig. 46. 













ELEMENTS OF ACOUSTICS. 


87 


describe the arc 0II' O '; this will represent a section, Construction of 
by the same plane, of the reflected pnlse. Draw any and 
incident ray as D G\ through the point given by the reflected pulses, 
value of f in Equation (32), and the point (7, draw IIC , 
which being produced will give the refracted ray G X '; 
through D' draw the line D' C X, and multiply the inter¬ 
cepted portion GXby the ratio of the velocities V and V, 
and lay off the product from G to X ', and we have the 
point X of the refracted, corresponding to the point X 
of the reflected pulse. An ear situated at X will hear 
the direct sound transmitted along the ray D X J and an Position whence 
echo of the same sound reflected at the point G: the the directsound 

. . to t • . and the echo are 

interval of time, or number ot seconds intervening be-both audible; 
tween the two, being equal to 


_ DC +CX-DX _ 

1089,42 %/ 1 + («° - 32°) . 0,00208 ’ 


Time between 
the impressions; 


on the supposition that the sound is transmitted through Position whence 
the atmosphere, and the linear distances are estimated in the transmitted 
English feet. An ear situated at X will hear the trans- sound 1S healcL 
mitted sound at the instant the one at X will receive 
the echo. 


§79. An echo is always produced when the ear is When an echo * 
able to distinguish the direct sound from that which is produced; 
reflected. A good ear will perceive about nine succes¬ 
sive sounds in one second of time; that is to say, the 
sounds must succeed each other at intervals of one-ninth Powers of the 
of a second in order to be heard singly. The sound and ear ’ 
the echo are to be regarded as successive sounds, of 
which the latter will be distinctly heard if it fall upon 
the ear after this organ has conveyed to the mind a dis- Time between * 

0 d > sound and its 

tinct impression of the former. The interval of time echo; 
between the sound and its echo, depends upon the dif¬ 
ference of route travelled by the direct and reflected - a 

sound, and the least difference x for a distinct echo, will • ’ 

result from the Equation J 





88 


NATURAL PHILOSOPHY. 


Least difference 
of route for a 
distinct echo; 


I s X 

9“ “ 1089,42 VT+{t° - 32°) 0,00208 ’ 


or, taking the temperature of the air at 32°, 


Same at 32°. 


X = 


1089,42 
9 


= 121,04. 


Sound and echo 

distinctly 

perceptible; 


Echo causes 
oonfusion; 


Effect of sound 
strengthened by 
echo; 


When the difference of routes exceeds this distance, the 
interval of time between the two impressions upon the 
ear becomes distinctly perceptible; and in proportion as 
that difference becomes less than a?, will the impression of 
the echo begin before that of the direct sound ends; and 
this overlapping, as it were, of impressions will give rise 
to confusion, which will continue to a greater or less ex¬ 
tent till the difference of routes becomes so small as to 
afford no sensible interval between the instants that mark 
the beginning of both impressions, in which case the echo 
will strengthen the effect of the direct sound. 


Illustration; 


Person assuming 
different 
positions 
between two 
walls; 


Fig. 47. 


o 


§ 80. Let an observer place him¬ 
self at O , midway between the plane 
walls A B and O J9, of which the 
distance apart is some 250 feet or 
more. The sounds which he utters 
will be reflected back to him by the 
two walls, and having traversed equal 
distances will reach him at the same 
instant; they will, therefore, rein¬ 
force each other, and he will hear 
one distinct echo. Now let him move towards one of the 
walls. At first he will perceive little or no difference of 
effect, but presently one echo will seem to lag behind the 
other, confusion will soon follow, and this will continue 
till twice the difference of his distances from the two 


walls becomes equal to or greater than 121 feet, when he 
will hear two distinct echos, which will separate more and 








ELEMENTS OF ACOUSTICS. 


89 


more from each other as he progresses; when he gets Effects observed, 
within sixty feet of the nearest wall, the first echo will 
begin to confound itself with the sound of his voice heard 
directly; he will now enter a second space of indistinct¬ 
ness, from which he will emerge at a distance from the 
wall of about fifteen or twenty feet. 


§ 81 . It thus appears that reflecting surfaces situated at surfaces at 
different distances from a speaker may throw back to him dis t an ce S reflect 
numerous echos of the same sound. Of this many re- man y echos of 
markable instances are recorded. At Lurley-Fels, on the 8amesoun ’ 
Rhine, is a position in which a sound is repeated by echo 
seventeen times. At the Villa Simonetta, near Milan, is 
another where it is repeated thirty times. An echo in a 
building at Pavia used to answer a question by repeat¬ 
ing its last syllable thirty times. The rolling of 
thunder has been attributed to echos from clouds situated 
at unequal distances from an auditor; and the propriety 
of this view has been sustained by the observations of SeTeral 
Arago, Matthieu and Proney, while experimenting upon instances; 
the velocity of sound. They found that when the weather 
was perfectly clear the reports of their guns were always 
single and sharp; whereas when the sky was overcast or 
a single cloud of any extent was present, they were fre- Experiments; 
quently accompanied with a long continued roll like that 
of thunder, and occasionally a double sound would arrive 
from a single shot. 


But it is proper to remark that the rolling of thunder 
admits of another explanation. Thunder is caused by a 
disturbance of electrical equilibrium in the atmosphere; 
experience shows that this takes place over a long and 
sinuous line, the different points of which are at unequal Roiling of 
distances from the auditor, and the sounds from these thunder - 
points can, therefore, only reach him in succession and 
without sensible intervals. 


§ 82. When reflected sound and that proceeding di¬ 
rectly from the same source, are made to fall upon the 



90 


NATURAL PHILOSOPHY. 


Reflected sound 
may increase the 
effect of direct 
sound; 


Illustrated by 
the speaking 
trumpet; 


Fig. 48. 





Its construciion 
and use 
explained; 


By its 
are rendered 
audible that 
could not be 
heard without it. 


ear simultaneously, or nearly so, they strengthen each 
other and become audible in positions whero neither 
could be heard separately. The /Speaking Trumpet 
affords an illustration of this. The Speaking Trumpet 
is a funnel-shaped tube, of which the object is to throw 
the voice beyond its ordinary range. In its best form 
it is parabolic. 

It is a geometrical property 
of the parabola that a line 
FT, drawn from the focus 
F\ to any point T of the 
curve, and another T K, 
drawn from T parallel to the 
axis FA, make equal angles 
with the tangent line to the 
curve at T. A portion of 

the diverging rays of sounds proceeding from a mouth 
at the focus F, will be reflected by the trumpet in 
directions parallel to the axis A F / and the living forces 
of the aerial molecules which, without the trumpet, 
would have been diffused over that portion of the spheri¬ 
cal surface on the outside of a cone of which FR and 
FR are the most diverging elements, become, by its 
use, concentrated within the limits of a circle whose 
diameter M H, is' equal to that of the trumpet’s mouth, 
and superposed upon the living forces arising from the 
action of the direct sound. The axis of the trumpet be¬ 
ing directed upon a person at a distance, sounds of audi-^ 
ble intensity may thus be conveyed to him, which he 
could not hear from the unassisted organs of speech. 


§ 83. The Hearing Trumpet , 
which is intended to assist per¬ 
sons who are hard of hearing, 
is similar to the speaking trum- 

Hearing trumpet; P e ^ 5 but the operation is re- 
versed. The rays of sound en¬ 
ter this instrument at the larger 


Fig. 49. 




















ELEMENTS OF ACOUSTICS, 


91 . 


opening and are so reflected as to become united at the Construction 
smaller end, which is inserted into the ear. 

§ 84. Whispering Galleries , so called from the fact whispering 
that the faintest whisper uttered at one point may be dis- sallenes ’ 
tinctly heard at another and distant point, without its be¬ 
ing audible at intermediate positions, depend upon the 
operation of the same principle, to wit, the convergence 
of the rays of sound by reflexion. The best form for Best fonn - 
these galleries is that of the ellipsoid of revolution. In 
such a chamber two persons, one in either focus, could 
keep up a conversation with each other which would 
be inaudible at other points. The ear of Dionysius is Ear of Dionysius, 
celebrated in ancient history; it was a grotto cut out * 
of the solid rock at Syracuse, in which a person placed 
at one point could hear every word, however faintly 
uttered, in the grotto. It was doubtless of a parabolic 
shape. 

The same principle is employed in 
the construction of Speaking Tubes , used 
for the purpose of communicating between 
different apartments of the same building, 
now coming into very general use. 

§ 85. Halls for public speaking, such as 
lecture rooms, theatres, churches, and the 
like, should be so constructed as to diffuse 
the sounds that are uttered throughout the 
space occupied by the audience, unimpaired 
by any echo or resound. Were the speaker 
to occupy constantly the same position, the 
parabolic form would,on theoretical grounds, 
undoubtedly be the best; but in debating 
halls, where every speaker occupies a dif¬ 
ferent position from another, these conditions are very Principles on 
difficult to fulfil, especially when the room is large. Every- 
thing should be avoided that would at all interfere with constructed. 


Fig. 50. 


Speaking tubes 
on same 
principle. 


Halls for public 
speaking; 






02 


NATURAL PHILOSOPHY. 


Experiment; 


Illustration; 


Explanation. 


the uniform diffusion of sound, and especially all need¬ 
less hollows and projections which are likely to* gene¬ 
rate echos. 

The following experiment will illustrate, in a very 
simple manner, the consequences arising from the re¬ 
flexion of the rays of sound from the interior of a pa¬ 
rabola. 

Place a watch in the focus A. of a parabolic mirror AT N , 
and all the rays of 
sound that fall on the 
concave surface will be 
reflected in the direc¬ 
tion indicated by the 
arrows. The ticking 
of the watch will be 
plainly heard within the space MNOP , in which the 
rays fall, but it will not be audible at a small distance 
on either side. 

Now place a second reflector 0 P, opposite to the for¬ 
mer, and at some distance from it; the rays of sound 
will be received by it and thrown into the focus B. If 
the ear, or, better still, the mouth of a hearing-trumpet, 
be applied to this point, the ticking of the watch will 
be heard as plainly as at A. 


Fig. 51. 



AT P 


§ 86. T^hile it is important to diffuse sound uttered 
Partition walls; or in any way produced, uniformly, so as to render it 
distinctly and equally audible in all directions, it is 
also necessary to prevent its passage from one apart¬ 
ment to another for which it was not intended. Parti¬ 
tions are usually made of solids; but solids, if elastic, 
such as wood, metals, and stone, are, as we have seen, 
better adapted to transmit sound than air itself; an 
essential condition, however, for this transmission is homoge¬ 
neousness of substance and uniformity of structure. Where 
now constructed these are wanting a sonorous pulse transmitted through 
transmission* of a so ^ ever changing its medium, and soon becomes 
sound. broken up by reflexion and refraction, retardation and 









ELEMENTS OF ACOUSTICS. 


93 


acceleration, into a multitude of non-coincident waves, Examples of 
and these from the laws of interference must, to a greater so^ erence ° f 
or less extent, destroy each other. 

As an instructive instance of this stifling effect on a 
sonorous pulse, we may mention the example afforded 
jby a tall glass filled with champagne. As long as the 
effervescence lasts and the wine is full of bubbles, the 
glass cannot be made to ring by a stroke on its edge, 
but will give a dead and puffy sound. As the effer-Glass of 
vescence subsides the tone becomes clearer, and when champagne; 
the liquid is perfectly tranquil, the glass rings as 
usual. On re-exciting the bubbles by agitation, the 
musical tone again disappears 

So of a solid or union of several solids, in which Heterogeneous 
there are frequent changes of density and elasticity, solids 5 
and especially where there is a want of adhesion among 
the different parts; sound penetrates these with great 
difficulty, and materials so united as to satisfy to the 
greatest extent possible the condition of non-homogeneous- 
ness should, therefore, be employed whenever it is an object 
to prevent the transmission of sound. The influence of 
carpets, curtains, and tapestry hangings, in preventing 
reflexion and echos in large apartments, is due to the 
causes above mentioned. The mixture of the unelastic carpets, curtains, 
fibres of the cloth with its numerous layers of entangled &c ' 
air, intercepts and deadens the sonorous waves before 
they reach the more solid and elastic media behind. 






94 


NATURAL PHILOSOPHY. 


Audible sounds 
produced; 


Impression on 
the ear depends 
upon; 


Auditory nerves 
analyze 
pulsations: 
whence grave, 
acute, &c. sounds; 
and tones of 
musical 
instruments. 

Noise; 


Crack; 


MUSICAL SOUNDS. 

§ 87. Every impulse mechanically communicated to 
the air or other elastic medium is, as we have seen, 
propagated onward in a wave or pulse; hut in order 
that it may affect the ear as an audible sound, a cer¬ 
tain force and suddenness are necessary. The slow weav¬ 
ing of the hand through the air is noiseless, but the 
sudden displacement and collapse of a portion of that 
medium by the lash of a w T hip, produces the effect of 
an explosion. The impression conveyed to the ear will 
depend upon the nature and law of the original impulse, 
which being altogether arbitrary in duration, violence 
and character, will account for all the variety observed 
in the continuance, loudness and quality of sound. The 
auditory nerves, by a most refined delicacy of mechan¬ 
ism, appear capable of analyzing every pulsation, and of 
appreciating the laws w r hich regulate the motions of the 
molecules of air in contact with the ear; and from this 
arise all the qualities—grave, acute, harsh, soft, mellow, 
and nameless other peculiarities which we distinguish 
between the voices of different individuals and different 
animals, and the tones of different musical instruments 
—bells, flutes, cords, &c. 

§ 88. Every irregular impulse communicated to the 
air produces what we call noise, in contradistinction to 
musical sound. If the impulse be short and single, we 
hear a crack; and as a proof of the extreme sensibility 
of the ear, it is to be remarked that the most short and 
sudden noise has its peculiar character. The crack of a 
whip, the blow of a hammer against a stone, the explo¬ 
sion of a pistol, are perfectly distinguishable from each 
other. If the impulse be of sensible duration and irre¬ 
gular, we hear a crash; if long and interrupted, a rattle, 


Crash; 



ELEMENTS OF ACOUSTICS. 


95 


or a rumble, according as its parts are less or more con- R imbie. ♦ 
tinuous. 

§ 89. The ear retains for a portion of time after the Continuou9 
impulse is communicated to it a perception of excitement, sound produced; 
If, therefore, a short and sudden impulse be repeated 
beyond a certain degree of quickness, the ear loses the 
intervals of silence and the sound appears continuous. 

The probable frequency of repetition necessary for the 
production of continuous sound is stated to be not less 
than sixteen times in a second, though the limit will be Th< fr ^" enc * v of 
different for different ears. necessary. 

§ 90. If a succession of impulses occur at exactly equal Musical sounds; * 
intervals of time, and if all the impulses be exactly simi¬ 
lar in duration, intensity, and law, the sound produced 
is perfectly uniform and sustained, and takes that pecu¬ 
liar and pleasing character called musical. In musical 
sounds there are three principal points of distinction, 
viz.: the pitch, the intensity, and the quality. Of these ^dquaiuy•^ 
the pitch depends, as we have seen, solely upon the fre¬ 
quency of the repetition of the impulses; the intensity, 
on their violence; and the quality, on the peculiar laws 
which regulate the molecular motions in any particu¬ 
lar instance. All sounds, whatever be their intensity or soundshaving 
quality, in which the elementary impulses occur with pitch ’ or 1B 
the same frequency, have to the ear the same pitch, and 
are said to be in unison. It is on the pitch alone that 
the whole doctrine of harmonics is founded- 

§91. The means by which a series of equidistant im- . , 
pulses can be produced mechanically upon the air are mechanically 
very various. If a toothed wheel be made to turn with produced; 
a uniform motion while a steel or other spring is held 
against its circumference with a constant pressure, each 
tooth as it passes will receive an equal blow from the 
spring, and this being communicated to the air, a wave 
of sound will proceed from the place of collision. The 



96 


Tho siren; 


A series of 
palisades; 


■Whistling of a 
bullet 


Most ordinary 
way of causing 
musical sounds; 


Modes 

considered. 


NATURAL PHILOSOPHY. 


number of such blows in a second will be known when 
the angular velocity of the wheel and the number of 
teeth upon its circumference are known, and thus every 
pitch may be identified with the number of impulses 
which produce it. The Siren, another instrument by 
which the same results may be evolved, has been de¬ 
scribed in § 48. A series of broad palisades, placed 
edgewise in a line running from the ear, and equidistant 
from each other, will reflect the sound of a blow struck 
at the end nearest the auditor, producing a succession 
of echos which reach the ear at equal intervals of 
time, thus producing a musical note whose pitch will be 
determined by the number of reflexions in each second 
of time. This number will be equal to the quotient 
arising from dividing the velocity of sound by twice the 
distance between two adjacent palisades. A similar ac¬ 
count may be given of the singing sound produced by a 
bullet while moving through the air and turning rapidly 
about its centre of inertia. The angular motion of the 
bullet being uniform, the actual velocity of its surface 
on one side will be greater than that on the other, and 
any inequality in the figure of the bullet will be made to 
vary its action upon the air periodically, thus producing 
a musical sound. 

§ 92. The most ordinary way of producing musical 
sounds is to set in vibration elastic bodies, as stretched 
strings and membranes, steel springs, bells, glass, co¬ 
lumns of air in pipes, &c., &c. All such vibrations con¬ 
sist in a regular alternate motion to and fro of the 
molecules of the vibrating body, and are performed in 
strictly equal portions of time. They are, therefore, 
adapted to produce musical sounds by communicating 
that regularly periodic initial impulse to the aerial mole¬ 
cules in contact with them, from which such sounds re¬ 
sult. We proceed to consider their modes of production, 
and especially in the first and last named cases, these 
being the most simple. 



ELEMENTS OF ACOUSTICS. 


97 


VIBRATIONS OF MUSICAL STRINGS. 


§ 93. If a string or wire be stretched between two vibrations of 
fixed points, and then struck or drawn aside from its musicalstrin s s i 
position of rest and suddenly abandoned, it will vibrate 
to and fro till its own rigidity and the resistance of the 
air bring it to rest; but if a fiddle bow be drawn across 
it, the vibrations will be renewed and may be maintained 
for any length of time, and a musical sound will be heard 
whose pitch will depend upon the greater or less ra¬ 
pidity of the vibrations. 

Thus, if MN be 
any stretched cord, 


struck 


at right an- 


M- 


Fig. 52. 

0 


A"' 


/S„ 


Illastratioa: 


gles to its length at 0 , 

it will be suddenly bent at that point into the curved or 
waved shape indicated by the dotted line/S, which shape will 
run alonff the cord in both directions till it meets with 

o 


some obstruction to its further progress, when it will be 

either wholly or partly reflected, and return upon its 

course in a manner and for the reasons to be explained wave runs along 

presently, the successive positions in the diverging mo- ^^g“ boUl 

tion being S\ /S'", &c., on the one side, and /S y , /S„, &c., 

on the other. 


§ 94. To find the velocity with which the wave runs t© find the 
alone: the cord, it is plain that we may either regard velocityof thi ® 

° x , ° , wave motion; 

the cord as. continuous, or as being composed of a senes 
of detached points, kept in relative position by their mu¬ 
tual attractions for each other, each point being loaded 
with the mass of so much of the cord as extends half 
way on either side to the adjacent point, and of which 
the length is equal to the distance between any two con¬ 
secutive points. 




98 


NATURAL PHILOSOPHY. 


Ill u&t ration; 


Suppositions; 


Consequences; 


Ye.ocity of a 
point of the cord; 


Tensions of parts 
of the cord; 


Fig. 53. 


& 

E B^-" 1 

i--ivr 


Suppose MN to be 
the cord’s position of 
rest, and the wave to 

proceed in the direc- M -jp —jj a m 

tion from 0 to D; let 

the point A , be just on the eve of motion and the place 
B\ the position of the point B at the same instant. 
While, therefore, the actual motion of the point B has 
been from B to B\ that of disturbance has been from 
B to A. 

The duration of these simultaneous motions is indefi¬ 
nitely short; the motions themselves may, therefore, be 
regarded as uniform. Hence, denoting the actual velo¬ 
city of the point B by v, and the velocity of the dis¬ 
turbance by V, we have 

v: V:: B B r : A B. 


v = V. 


BB' 

AB 


or, denoting the angle BAB' by 9 , in which case, 

BB' 


A B 


= tan 9 , 


we find, 


v = V. tan 9 .( 33 ). 

The tension of the cord between A and B , acts to 
draw the point A , from A towards B, and the tension 
between A and D acts to draw the same point from A 
towards D. Denote the tension of the cord when at rest 
by C ’, that between A and B' by G\ then because the ten¬ 
sion of the same portion of the cord will be proportional 
to the length to which it is stretched, will 







ELEMENTS OF ACOUSTICS. 


00 


whence 


C : G'wAB'.AB’ 
AB ' 


Relation of 
tensions 0 and 
C‘; 


O' = ( 7 . 




and this being resolved into two components, one acting 
from A to E, at right angles to MN^ the other in the 
direction of M iT, will give for the first 

C\ sin 9 = O'. = C. = C. tan 9, 


A B' 


AB 


Components of 
C; 


C. cos ? = O'. rUJ = C; 


the second will be destroyed by the tension from A to 0ne is destroyed 
Z?, while the first will alone produce motion in A , and the motive force; 
is, therefore, the motive force. Denote by w , the weight 
of a unit’s length of the cord while at rest, then will 
the mass with which A is loaded be expressed by 

Mass on which 
the motive force 
acts; 

in which g denotes the force of gravity; and the accele¬ 
ration due to the motive force will be 


AB, 

9 


C . q. tan 9 
w. AB' 


Acceleration due 
to the motive 
force; 


and therefore the velocity of A, in the small time t, which 
velocity will be equal to that of B’ when JD begins to 
move, will be given by the relation 


v = —? . tan 9. 


w 


t 

AB * 


t _ 

AB~ V ’ 


But 


Velocity of a 
particle in small 
time i; 










100 


NATURAL PHILOSOPHY. 


and denoting by I /5 half the length of the cord whose 
weight is equal to the tension (7, we have 


Value of tension 
<7/ 

C = 2 W. I t 


which values substituted above give 

Velocity of 
particle in small 
time t. 

2 g L r tan 9 

V 9 


and replacing tan 9 by its value found from Equation 
(33), gives 

V 2 

or 


Velocity of way* V — V 2 g L { .(34). 

along a cord; 

Rule. That is to say, the velocity with which a wave or pulse 

will run along a tense cord is constant , and equal to 
that acquired by a heavy body in falling in vacuo , un¬ 
der the action of its own weight , through a height equal 
to half the length of the cord whose weight*is equal to 
the tension. 

Example; Example. A cotton thread 73 feet long and weighing 

904 grains, is stretched by a weight of 12840 grains; 
with what velocity will a wave move along this cord ? 

Eirst 


904 : 12840 :: 73 : 2 1, 

whence 


Computation* 


2 L, = — 12840 = 1036^83. 


904 







ELEMENTS OF ACOUSTICS. 


101 


Second 

V = s/TaTI, = V 32,18.1036,83 = 182^4. Wave velocity 

' 7 along the cord; 

§ 95. Substituting the above value for V, in Equation 
(33), the latter becomes 


v = V 2 q L r tan <p, . . . . (35) 

^ ; 5 ' ' Velocity of a 

point of the cor 4; 

from which it appears that the actual velocity of a point 
of the cord is directly proportional to the tangent of the 
inclination of the cord, at that point, to the cord’s posi¬ 
tion of rest; and when this condition ceases to obtain, 
as it does when the pulse comes to lighter and more 
movable portions, or encounters obstacles less mova¬ 
ble than the rest of the cord, it will be divided into Pulsemoyin & 

along a cord 

two, one of which will continue to move in the same resolved into 
direction while the other will run back, or be reflected, two; 
and produce a kind of echo, just as in the case of a 
wave of air encountering a medium whose molecules are ‘ 
either more movable or less so than those of air. 


If one end of the 


Pier 



cord be fixed at Jf, its 


JC lg. O -*• 


One end of the 

molecules adjacent to 

M — 

A ^ As 

- 'N 

cord fixed; 

those in contact with 





the fixed obstacle tend¬ 

w— 




ing, when the pulse 





reaches the latter, to 

My— 




move at right angles 




to the cord’s length, 

My— 

- 

■V _ s—m 


will be resisted by the 





stationary molecules; 

Mt— 


- - uvr 


the reaction will throw 





them to the opposite 

3ft— 

-- y 

- |2V 


side, and this reaction 




Pulse thrown to 
the opposite side 

extending to the mole¬ 




of the cord and 

cules behind, the pulse 

will 

pass to the oppoite side of returns; 

















m 


NATURAL PHILOSOPHY. 


Both ends of the 
cord fixed; 


Eefiected pulses 
meet and 
conspire; 


Separate and are 
again reflected; 


Fig. 55. 


M t- 


. 2 V 


Meet a second 
time at the point 
of primitive 
disturbance; 


Cord brought to 
rest. 


the cord and return along its entire length, following 
after the direct pulse in the same direction. If the 
second end be fixed, the direct pulse proceeding towards 
it will conduct itself in the same way; the reflected pulses 
will proceed to meet each other, and being on the same 
side of the cord will conspire at their place of union to 
produce a single resultant pulse, in which the molecules 
of the cord will depart from their places of rest by the 
sum of the distances of the same molecules in the com¬ 
ponent pulses. These component pulses will, however, 
immediately separate, 
and proceed towards 
the fixed ends, where 
they will be reflected 
as before, and return 
to meet again, having 
once more changed 
sides. The point of 
second meeting will be 

at the place of primitive disturbance, from which the 
waves will depart, as before, to undergo the same round ; 
and thus, but for the resistance of the air and want of 
perfect elasticity in the cord, the latter would vibrate 
for ever. But every pulse communicated to the air, is 
an elimination from the cord of so’ much of its living 
force , and as this must soon become exhausted, the cord 
will come to rest. 


jfh 


■LT 




ar* 


r. 56. 


-dr 


§ 96. Suppose the 
whole length of the 

Whole length of cor d Jj£ N to be de- ^ 2- 2' 

cord divided intx> , , T 7 ,, ~~o 

two parts; noted by L = l t + l , 
of which l t represents 

the distance 0 M, from the point of primitive disturb¬ 
ance (?, to the fixed obstacle on one side, and l r the dis¬ 
tance ON to the obstacle on the opposite side. Then 
denoting the time of describing V by t\ and that of de¬ 
scribing l t by t { , we have 








ELEMENTS OF ACOUSTICS. 


103 



V = V. 

Lengths of the 


ii 

parts; 

whence 

II 

^1^ 

Times required 
for a pulse to pass 
along them. 

t - A_ , 



the first pulse being reflected at JV, will describe the 
entire length V + l t in the reverse direction in the time 


t' +t l = 


il + A 

F V 


Time in which 
the first pulse 
describes the 
whole length; 


and the second pulse being reflected at M, will describe 
the entire length V + l n in the same time, or 


t + 1= 



The same for the 
second pulse ; 


the first pulse being reflected a second time at will 
describe the length l, in the time 

Time in which 
first pulse passes 
over second part-; 



and the second pulse being reflected a second time at iV, 
will describe the distance l\ in the time 



That in which 
second pulse 
passes over first 
part; 


and at the expiration of all these times the pulses will 



104 


NATURAL PHILOSOPHY. 


Pulses return to be at their first starting point, and each having been 
starting point; twice reflected, they will be on the same ’side of the 
cord that they were originally ; they will, therefore, pro¬ 
duce a resultant pulse precisely the same, abating the 
qualification due to the air and imperfect elasticity, as 
that produced by the initial impulse. Hence, if T de¬ 
note the time of one complete vibration of the cord, that 
conspire and is to say, the interval between the instant of primitive 
resSt^tpuise; disturbance and that at which the cord resumes its in¬ 
itial condition, we shall have, by taking the half sum 
of these several intervals—because both pulses are mov¬ 
ing during the same time, 

Time of vibration T z=z% it' + t) ^ ^/) ^ 

of the cord; V '' ~ y ~ V 

and replacing V by its value, Equation (34), 


The same 
reduced. 


2 L 


. . . (36) 


Exam ie- Example. Taking the example of § 94, in which L = 73, 

and 2 L t = 1036,83 feet, we find 

Result T — — — - _ 0*799. 

V 32,18 .1036,83 


§ 97. The truth of the foregoing theory has been fully 
confirmed by the experiments of Weber. He stretched 
a very uniform and flexible cotton thread fifty-one feet two 
inches in length, weighing 864 grains, horizontally, by a 
Experiments ; known weight. The thread was struck at six inches from 
the end, and the time of the wave’s running a certain 
number of times over the length of the string, back¬ 
ward and forward, carefully noted by means of a stop¬ 
watch that marked thirds (the sixtieth part of a second). 








ELEMENTS OF ACOUSTICS. 


105 


The mean of a great many trials, agreeing well with Mean of result* 
each other, gave the results in the following table: 


Tension in 
grains. 

Length run over 
by the wave. 

Time in thirds. 

Time of running 
over the length 

10^,33 in thirds 
by observation. 

Time by calcu¬ 
lation from the 
formula 

T- 

Vzql, 

10023 

102^4 

46 

46 

46,012 

10023 

204,7 

92 

46 

46,012 

10023 

409,4 

184 

46 

46,012 

33292 

409,4 

99 

24,72 

25,246 

69408 

409,4 

65 

16,25 

17,485 


A more complete confirmation could not have been 
desired. The slight discrepancies are doubtless owing 
to a want of perfect uniformity in so long a thread, 
which must necessarily have formed a catenary of sen¬ 
sible curvature. 

Denote by N the number of vibrations performed in 
a given time T J, then will 

Time of one 
vibration of the 
cord; 



which substituted for T in Equation (36) gives, after 
taking the reciprocal of both members, 


N _ V 2 g L, 
2 1 


(3Y). 


Keciprocal of the 
same. 


In the foregoing equations 2 L n denotes the length of 
the cord of which the weight measures the tension. 

Denote this weight by IF, the diameter of the cord by 
D y and its density by d / then will 

Weight of cord 
whose length 
measures the 
tension; 


W = 7T. ^ .2 L,.d.g 













106 


NATURAL PHILOSOPHY. 


Length of half 
this cord; 


Time of 
vibration; 


Its reciprocal; 


Rule first; 


Rule second. 


Vibrating cord 
fixed at both 
ends and struck 
ir + he middle; 


Primitive pulse 
resolved into 
two; 


whence 


T 2 TF 

' “ 57 l) 2 .d.g 


which substituted in Equations (36) and (37), give 


T= V* 


D. L. Vd 

7TF~ 


. . (38) 


JT _ 1 VW 

' D.L.Vd' 


. . (39) 


from which it appears that, the time of vibration of a 
tense cord varies as its length , diameter and square root 
of its density , directly ; and the square root of the stretch¬ 
ing force , inversely. And that; the number of vibrations 
performed by a tense cord in a given time , varies as the 
square root of the stretching force directly , and the diame¬ 
ter , length and square root of the density inversely. 

§ 98. The tense 

® Fig. 57. 

cord MW being fixed 

at both ends and in a M \ - ' —- & 

state of vibration, ap¬ 
ply the finger, or any other partially obstructing cause, 
at the middle point F\ and then withdraw it. The law 
of Equation (34), will be suddenly interrupted at this 
point, the progressing pulse will be resolved into two, 
one of which will continue to move in the same direc¬ 
tion and on the same side of the cord, while the other 
will be reflected and return along the opposite side. 
These component. pulses having equal distances to tra¬ 
vel before they reach the ends, will be reflected at the 
fixed points at the same instant, return on opposite 
sides of the cord, and meet in the centre. They will, 







ELEMENTS OF ACOUSTICS. 


107 


therefore, solicit simul¬ 
taneously the central 
point 0 , in opposite di¬ 
rections, and if the 
pulses he equal, they 
will wholly interfere at 


Fig. 58. 



The two 
reflected pulses 
interfere at the 
middle of the 
W cord; 


that point, which must, therefore, remain stationary. The 
effect of the reciprocal action of the two waves being to fix the 
point 0, these waves will both be totally reflected there, 
will return to the ends, be again reflected, return to the Are thence a?afa 

cy 1-11 reflected and so 

centre, from which they will be thrown back towards on. 
the ends, and so on till the living force of the cord is 
totally expended upon the air. Thus the two portions 
M 0 and 0 N of the cord may vibrate as though the 
point 0 had been originally fixed. 


If the finger be ap¬ 
plied but for an in¬ 
stant at 0 , at a dis¬ 

M\ — 

Fig; 59. Finger applied at 

one third the 

^ whole length 

tance from M equal 
to one-third of the 

J£\ _ 

^ 77 — * --- —from the end; 

whole length M iV 7 *, 
while the wave is pro¬ 
gressing from M to¬ 
wards 0 , the latter 

M \— 

- ^ ^ M 

_ O Ar 


- 


will be resolved, as before, into two component waves, rrimitive pulse 
one of which will continue to move towards JV on the ™ s ° hed int0 
same side of the cord, the other will return to M on the 
opposite side. The distance MO being equal to one- 
half of 0 JSf, the return component will be wholly re¬ 
flected and change sides at M\ and come back to the 
point 0 by the time the direct component arrives at 
where the latter will be totally reflected and pass to 
the opposite side of the cord. The component waves 
being now on opposite sides of the cord, and moving 
towards each other, one starting from 0 and the other 
from iT, will meet at 0\ half way from O to iT, component 
making JV O' equal to one-third of M JV. Here P ul8esMerfare » 
they will interfere, be totally reflected, and proceed from 








108 


NATURAL PHILOSOPHY. 


Are again 
reflected; 


Again meet at 
their starting 
point and so on. 


Finger kept on 
the cord; 


One reflected 
component 
resolved into 

two; 


Reciprocal 
action between 
these two sets of 
components; 


Cord broken up 
into portions, 
each one 
vibrating. 


O' as they did from 0; 
they will meet again in 
this latter point and 
there be totally reflect¬ 
ed, and thus each com¬ 
ponent wave will be 
made to describe, as 
long as the cord 
retains any of its 
living force, alternately one-third on one side and two 
thirds on the opposite side of the entire length of the 
cord, as though the point 0 were to become alternately 
fixed at 0 and 0\ after every reflexion at M and N. 

Were the finger to 

be kept at the point 0 , rig. eo. * 

till the first reflected 0 

component returned to o' 

that point, this compo¬ 
nent would be there subdivided; giving rise to a second 
return as well as a second onward component; the lat¬ 
ter would meet the first onward component at 0\ and 
by its action upon it resolve this also into two com¬ 
ponents, the onward one of which would meet the 
second return component at 0 , and being on opposite 
sides would interfere and hold this point at rest. Thus 
the whole cord may be broken up into three equal 
parts, each of which will vibrate as though the points 
of division, 0 and 0\ had been stationary or fixed. 

A similar explanation would show that if the finger 
were applied at any other point of which the distance 
from one of the fixed ends were an aliquot part of the 
whole length of the cord, the cord would in like man¬ 
ner be broken up, as it were, into equal aliquot portions, 
each of which would vibrate as though its extremities 
were fixed. 

Molecules or particles of a vibrating body thus ren¬ 
dered stationary by the simultaneous action of opposing 
waves or pulses, are called Nodal points. The interme- 


Fig, 59. 


M\- 

M\- 


O ' ~ > 


aX 


O 

o S -N 


O' 




Nodal points. 







ELEMENTS OF ACOUSTICS. 


109 


diate portions which, vibrate, are termed bellies, or ven- v en trai 
tral segments. segment*. 

§ 99. If L, denote, as before, the entire length of the 
cord, and n, the number of ventral segments into which 
it divides itself, then will the number of its nodes be 
— and the length of each segment, 

Length of a 
segment 

which substituted for L in Equation (36), gives for the 
time of vibration, 


L 


m 2 L 

rn __ _ 

n ' J YgL, 


(40) 


Time of 
5 vibration; 


and in Equation (37), the number of vibrations in the 
time T t , 


N VZg.L t 
T t ~ 2 L 

and for the number in one second, by making T { equal 
to one second, 


Number in time 
T,; 


Number in one 
* second. 

All of this is confirmed by experience. If the string Abovo 
of a violin, or violincello, while maintained in vibration deductions 

-it-it ii-i i confirmed by 

by the action ot the bow, be lightly touched by the experience; 
finger, or a feather, exactly in the middle or at one-third 
of its length, from either end, it will not cease to vibrate, 
but its vibrations will be diminished in extent and increased 
in frequency, and a note will become audible, more faint 
but more acute than the original, or fundamental note, 
as it is called, and corresponding, in the former case, to 


j\r = n ^ 2 y-A 
2 L 











110 


NATURAL PHILOSOPHY. 


illustrated by tiie a double, and in the latter, to a triple rapidity of vibra- 
vi° iin ; tion. The note heard in the first case being, in the 

scale of musical intervals, an eighth or octave, and in 
the second a twelfth, above the fundamental tone. If 
a small piece of paper cut in the form of an inverted Y, 
be set astride on the string, it will be violently agitated 
or thrown off if placed on the middle of a ventral seg¬ 
ment, but at the node will ride quietly as though the 
Harmonics. string were at rest. The sounds thus produced are termed 
Harmonics. 


Coexistence and 
superposition of 
small motions; 


Its application 
illustrated; 


Explanation; 


Results 
confirmed by 
experience. 


Fig. 61. 


§ 100. But further, according to the principle of the 
coexistence and superposition of small motions, referred 
to in § 56, any number of the various modes of vibra¬ 
tion of which a cord is susceptible, may be going on 
simultaneously. 

Thus, if we suppose a 
mode of vibration repre¬ 
sented by figure ( a ), in 
which there is no node, 
and another of the same 
cord represented by figure 
(J), with one node, to be 
going on at the same time, 
there will be a resultant 
vibration represented by 
the curve in figure (c), of 

which the ordinates are equal to the algebraic sum of the 
corresponding ordinates of the curves in figures (a) and 
(h). If a third mode of vibration, represented by figure 
(<#), be superposed upon the other two, there will arise a 
resultant vibration represented by the curve in figure (V), 
of which the ordinates will be equal to the algebraic 
sum of the corresponding ordinates in figures («), ( b ) and 
(d\ or, which is the same thing, the algebraic sum of the 
corresponding ordinates of figures (c) and id). 

This is also confirmed by experience. It was long 
known to musicians, that besides the fundamental note 











ELEMENTS OF ACOUSTICS. 


Ill 


of a string, an experienced ear could detect in its sound, Harmonic 
when in motion, especially when very lightly touched somuis; 
in certain points, other notes related to the fundamental 
one by fixed laws of harmony, and which are therefore 
called harmonic sounds. They are the very sounds that 
may be heard by the production of distinct nodes as ex¬ 
plained in § 99, and thus insulated as it were from the 
fundamental and other coexisting sounds. 

§ 101. The Monochord is an instrument adapted to ex- Themouochord; 
nibit these and other phenomena of vibrating strings. 

It consists of a single string of catgut or metallic wire 
stretched over two fixed and well defined edges to¬ 
wards its extremities, which effectually terminate its 
vibrations in the direction of its length; one end is 
permanently fixed, and to the other is attached a 
weight which determines the tension. The interval be-Essential part*; 
tween the two edges is graduated into aliquot parts, 
and the instrument is provided with a movable bridge 
or piece of wood capable of being placed at any 
point of the graduated scale, and abutting firmly against 
the string so as to stop its vibrations, and divide it into 
two equal or unequal parts, as the case may be. 

By the aid of this instrument may readily be found its use; 
the number of vibrations which corresponds to any given 
note of any particular instrument, as a piano-forte, for in¬ 
stance. To this end, it will only be necessary to know, 
when the note of the monochord is the same as that of the 
instrument, the distance L between the edges, the 
stretching weight, and the weight of a unit’s length 
of the string. The quotient obtained by dividing the 
former of these weights by the latter will give the va¬ 
lue of 2 Z,, in Equation (37), and making T t equal to one 
second in that Equation, we have for the solution of the 
question 


at _ ^ Z, 
2 L 


(42") Practical 

' / formula. 




112 


NATURAL PHILOSOPHY. 


Number of This gives the number of impulses made upon the ear 
ImTespondingto i n a secon d, corresponding to the fundamental note. To 
any higher note, obtain the number which answers to any note sharper, 
higher , or more acute , we have but to apply the bridge 
and slide it to some position such that the portion of 
the cord between it and one of the edges gives the 
note in question; the scale will make known Z, which 
in Equation (42), will give the number N. 


Harmonic tones 
produced by 
causing air in 
motion to 
impinge against a 
stretched cord. 


Two cords near 
together, and the 
shorter made to 
vibrate; 


Its vibrations 
will be 

transferred to the 
longer cord; 


§ 102. The contact of a stretched cord with solid sub¬ 
stances is not the only means of producing its fundamen¬ 
tal and harmonic tones. The sonorous pulses proceed¬ 
ing from a vibrating cord are but the consequences of 
repeated conflicts between the elastic force of the cord 
and that of the air. The former impresses upon the air 
a certain amount of living force, and the latter by its 
reaction transmits this living force through the atmos¬ 
phere to a distance. Reverse the process. Impress upon 
the air the same motion, and subject a stretched cord to 
its influence. Action and reaction only change names, 
and the cord must take up the motion of the air. Two 
cords equally stretched, and in all other respects similar, 
but the length of one only a half, a third, or any ali¬ 
quot part of the other, being placed side by side, and 
the shorter put in motion, the longer will soon assume 
a mode of vibration by which it will be divided into 
ventral segments, each equal to the length of the 
shorter cord. The sonorous pulses diverging from the 
shorter cord will arrive at the longer; and the mole¬ 
cules in the first of these pulses will, in their forward 
movement, press upon the stationary cord and give it a 
slight motion in their own direction. On the retreat of 
these molecules, the excess of aerial condensation will 
change to the opposite side of the cord; the latter will 
yield to the action of this inverted force and that of its 
own elasticity, and pass to some position on the oppo¬ 
site side of its place of rest, where being met by a second 
onward pulse, it will be thrown back in the direction of 



ELEMENTS OF ACOUSTICS. 


113 


its first motion, and thus made to undergo the same 
round as before. 

This process being repeated a number of times, the 

_ n 7 Synchronal 

cord will be set in full and audible vibration. But vibrations; 
these vibrations will obviously be synchronal with the 
aerial pulsations , and therefore , with the vibrations of 
the shorter cord , a condition that can only be fulfilled by r 

7 J Longer cord is in 

the longer cord breaking up, as it were, into portions effect broken up; 
of which the lengths are equal to the length of the 
shorter cord; for, the tension, diameter and density, of 
the cords being the same, the times can only be 
equal, Equation (38), wdien the vibrating lengths are equal. 

All motions of the longer cord which are inconsistent 
with this, though they may be excited for the moment 
by one pulsation, will be extinguished by the subsequent 
one. Hence, if two cords can have any mode of vibra¬ 
tion in common, that mode may be excited in either 
of them, and that only, by exciting it in the other. For 
example, if two cords, in all other respects alike, have 
lengths which are to each other in the proportion of 
2 to 3, and if either be set in motion, the mode of illustration, 
vibration corresponding to a division of the first into 
two and of the second into three ventral segments, will, 
if it exist in the one, be communicated by sympathy to 
the other. Indeed, if it do not originally exist, it will, 
after awhile establish itself; for, all the circumstances 
which may favor such a division, however minute, will 
have their effect preserved and continually accumulated, 
and thus become sensible. 

And it is important to remark that whether the primi¬ 
tive portion disturbed be large or small, whether it occu¬ 
py the whole string at once or run along it like a bulge ; 
whether it be a single curve, or composed of several ven¬ 
tral segments with intervening nodal points, we must not 
forget that the motion of a string with fixed ends is no sSlg wUhLod 
other than an undulation or pulse continually doubled back ends analogous 
upon itself , and retained within the limits of the cord reined within 
instead of running off both ways to infinity. certain limits. 

8 



114 


NATURAL PHILOSOPHY. 


Vibrations 
seldom confined 
to the same 
plane; 


Orbits described 
by particles may 
be observed; 


Specimens. 


Vibrating 
column of air of 
definite extent; 


§ 103. It is very seldom that the vibrations of a string 
can lie in the same plane. They most commonly consist 
of rotations more or less complicated, except when pro¬ 
duced by the sawing of a bow across the string. The 
actual orbit described by any one molecule may be made 
matter of ocular inspection by throwing the solar rays 
through a narrow slit so as to form a thin sheet of light. 
A polished wire stretched in such manner as to penetrate 
this sheet at right angles, will appear, when stationary, as 
a bright spot where it pierces the light, but when in mo¬ 
tion, the point of intersection will form a continued lumi¬ 
nous orbit, just as a live coal whirled round appears like a 
circle of fire. The figures exhibit specimens of such 
orbits observed by Dr. Young. 


Fig. 62. 



C3 



VIBRATING COLUMN OF AIR OF DEFINITE EXTENT. 

§ 104. The circumstances of the molecular vibrations 
of a stretched cord of indefinite extent, are, as we have 
seen, similar to those of a sounding column of air; and the 
tacts which have been stated respecting a vibrating cord 
are equally true of a vibrating column of air of definite 
extent. 






ELEMENTS OF ACOUSTICS. 


lltf 


Thus, if such a cylindrical 
column he enclosed in a pipe 
A B = Z, stopped at both 
ends by immovable stop¬ 
pers, and an impulse be com¬ 
municated in the direction 
C A) to one of its sections C, 
at the distance A C=l, from 
the end A, and B C=V, from the end B , this impulse 
will, § 69, give rise to two pulses running in opposite impulse 
directions. In the pulse from (7 to A the air will be con- com “ uni « atedto 
densed, and in that from C to B it will be rarefied, direction of the 
These pulses will be reflected at the stoppers, and the of th * 
condensed pulse, after passing over the distance l be¬ 
fore and l' after reflexion, will meet the rarefied pulse 
at the distance l from the end B , and produce a com¬ 
pound agitation in the section O' similar to that of the Two pulses will 
original disturbance ; thence the partial pulses will sepa- runti^fn 
rate, and after each undergoing another reflexion will opposite 
unite in their original point of departure, constituting, dliectl0U& ’ 
as it were, a repetition of the first impulse, and so on, Pulses 
till the pulses are destroyed by the gradual transmis- destroyed, 
sion of the whole of their living forces through the sub¬ 
stance of the tube to the open air. 

If the section first set in motion be maintained in a Consequence of 
state of vibration synchronous with the return of the Oration the 11 
reflected pulses, it will unite with and reinforce them at section first 
every return, and the result will be a clear and strong dlstulbe(L 
musical sound, resulting from the exact combination of 
the original periodic impulse with its echos. 


Tube closed at 

Fig. 63. both ends, 

__ containing air; 




A C 11 




AC’ B 


Fig. 64. 


§ 105. Let us suppose the 
section first set in motion and 
so maintained, to be exactly 
in the middle of the pipe. 

Then, when once the pe¬ 
riodic pulsation of the contained air is established, the 
motion will consist of a constant and regular fluctu- 


Middle section 
maintained in 
vibration; 









116 


NATURAL PHILOSOPHY. 


Air condensed in 
one half and 
rarefied in the 
other; 


Tositions of 
greatest and least 
condensations 
and rarefactions; 


Several columns 
end to end; 


Illustration; 


Nodes; 


Ventral 
segments; 


Distance between 
two alternate 
nodes. 


An opening in 
the middle, of a 
segment; 


ation to and fro of the whole mass, the air being always 
condensed within one-half of the pipe while it is rare¬ 
fied in the other. The greatest excursions from their 
places of rest will be made by the molecules in the 
middle, while the molecules at the ends abutting against 
the solid stoppers will have the least motion, the ex¬ 
cursion made by each intermediate molecule being 
greater in proportion as it is nearer the centre. On 
the other hand, the rarefactions and condensations are 
greatest at the extremities and diminish as we approach 
the middle, where they are the least. 

Now, conceive several such columns of vibrating 
air to be equal and to be placed end to end, so that 
the condensed portions shall be turned towards each 
other; it is plain that all the stoppers, except the ex¬ 
treme ones, may be removed without in anywise sen¬ 
sibly changing the interior motions, and there will re¬ 
sult a single column of air 
broken up into equal por¬ 
tions vibrating in a manner 
similar to that of the ventral 
segments of a tense cord, <— —* <— 

§ 98, the nodes being at X 

and X, where there will be alternately a maximum and 
minimum of condensation, the bellies lying between — 
in the middle of which the condensation will be the 
least. It is also obvious that the distance XX, be¬ 
tween two alternate nodes, will be the shortest dis¬ 
tance from any one section of air to another having 
the same phase, and that this distance answers to the 
length of a wave of the same pitch propagated in an 
indefinite column of air. 


Fig. 65. 
X Y 


§ 106. At 0, half way be¬ 
tween two consecutive nodes, 
or in the middle of one of 
the cylinders A B, let an 
opening be made; and sup- 


Fig. 66. 

X _ C _ T 









ELEMENTS OF ACOUSTICS. 


117 


Fig. 67. 


pose a vibrating body to be inserted whose vibrations 
are executed in equal times with those in which the 
excursions to and fro of the included aerial sections 
are performed in the stopped pipe. Its vibrations will 
be communicated to those of the contained air, the 
latter will be maintained and strengthened, and the 
sound from the pipe will become full and clear. Such 
an aperture is called an embouchure. 

Hext conceive one-lialf 
B (7, of the cylinder A B, 
to be removed, and in its 
place a disc substituted ex¬ 
actly closing the aperture, 

and maintained by some external cause in a state of 
constant vibration, such, that the performance of one 
complete vibration, going and returning, shall occupy 
as much time as a sonorous pulse would take to tra¬ 
verse the whole length of the stopped pipe A B , or 
double that required for the half pipe A C. Its first 
impulse on the air will be propagated along the half 
pipe C A, and reflected at the stopped end A, and will 
again reach the disc just as the latter is commencing 
its second impulse. But the absolute velocity of the 
disc in its vibrations being excessively minute compared 
with that of sound, the reflected pulse will undergo a 
second reflexion at the disc as though it were a fixed 
stopper. It will, therefore, in its return exactly coincide 
and conspire with the second impulse of the disc, and 
the same process being repeated at every impulse, each 
will be combined with all its echos, and a musical tone 
will be drawn from the pipe vastly superior to that 
which the disc vibrating alone in the open air could 
produce. This is the simplest instance of the resonance 
of a cavity. How, it is manifestly of no importance 
whether the pulses reflected from the closed end A of 
the semi-pipe undergo a second reflexion at the disc 
and are so turned back, or whether we regard the disc 
as penetrable by the pulse, and suppose the latter to 


And a vibrating 
body introduced; 


Embouchure. 


One half of the 
cylinder replaced 
by a vibrating 
disc; 


Reflected pulse 
will coincide 
with the second 
impulse of the 
disc, and so on. 


Resonance of a 
cavity. 




118 


NATURAL PHILOSOPHY. 


disc and be 
reflected at the 
other end. 


same effects mil run on anc j b e reflected at the extremity B of the other 
mlslhrough^the 6 half of the entire tube, and on its return again to pass 
freely through the disc and be again reflected at the 
end A. The sound will be the same on the principle of 
the superposition of vibrations. Thus the fundamental 
sound of a pipe open at one end is the same as that 
of a pipe closed at both ends and of double the length, 
and has the same pitch as that due to waves propa¬ 
gated in the open air, and of which the length of each is 
four times the length of the pipe open at one end. 


Eu-p Oriental 
ILiostration; 


Tuning fork and 
pipe; 


Flute male to 
speak. 



§ 107. The mode here supposed of exciting and sus 
tabling the vibrations of a column of air in an open tube 
may easily be put in practice. Take a common tuning- 
fork and by means 

of sealing wax fas- Flg ’ 68 ‘ 

ten a circular disc 
of card on one of its 
branches, sufficient¬ 
ly large to nearly co¬ 
ver the open end of 
a pipe. The upper 
joint of a flute with 
the mouth hole stop¬ 
ped will answer well 
for the purpose; it may be tuned in unison, that is, 
made of proper length by the sliding stopper. The 
fork being set in vibration by a blow on the unloaded 
branch, and held so as to bring the disc just over the 
mouth of the pipe, a note of great clearness and strength 
will be heard. Indeed, a flute may be made to “speak” 
perfectly well by holding a vibrating tuning-fork close 
to the embouchure, while the fingering proper to the 
note of the fork is at the same time performed. 



§ 108. But the most usual method of exciting the vibra¬ 
tions of a column of air in a pipe is by blowing across 
the open end, or across an opening made in the side 





















ELEMENTS OF ACOUSTICS. 


119 


or by introducing a current 
of air into it through a small 
aperture of a peculiar con¬ 
struction called a “reed” 
provided with a “ tongue ,” 
or flexible elastic plate which 
nearly stops the aperture, 
and which is alternately 
forced away by the current 
of air and brought back by 
its own elasticity, thus pro¬ 
ducing a continued and regularly periodic series of in¬ 
terruptions to the uniformity of the stream, and a sound 
in the pipe corresponding to their frequency. 

Except, however, the reed be so constructed as to be in conditions to be 
unison with some one of the possible modes of vibration fulfilIed ’ 
of the column of air in the pipe, the sound of the reed 
only will be heard, the resonance of the pipe will not be 
called into play, and the pipe will not speak; or will 
speak but feebly and imperfectly and yield a false tone. 

§ 109. Let us consider what takes place when the vi- Effect of blowing 
orations oi a column oi air are produced by blowmg en d 0 f a pipe; 
across the open end of a pipe or an aperture in the 
side. The current of air being so directed as to graze the 
opposite edge, a small portion will be caught and turned 
aside down the pipe, thus giving a first impulse to the 
contained air and propagating down it a pulse in which 
the air is slightly condensed. This will be reflected at 
the end as an echo and return to the aperture where the 
condensation will go off, the section condensed expanding 
into the free atmosphere. But in so doing it lifts up and 
for a moment diverts from its course the impinging cur¬ 
rent, and thus suspends its impulse upon the edge of the 
aperture. The moment the condensation has escaped P roduction 
the current resumes its lormer course and again exp]ained; 
touches the opposite edge, creates there a second conden¬ 
sation and propagates down the pipe another pulse, and 


Resonance of a 
pipe produced by 









120 


NATURAL PHILOSOPHY. 


Current 
alternately 
grazes and misses 
the edge. 


Point of 
maximum 
excursions of 
molecules; 


Vibrations of a 
column of air 
and of a cord; 


Cases made 
analogous. 


Can be no half 
segments in cords 
with fixed ends; 


Not so with 
pipes; 


so on. Thus the current passing over the end or aperture 
is kept in a constant state of fluttering agitation, alter¬ 
nately grazing and passing free of its edge at regular in¬ 
tervals equal to those in which the sonorous pulse can run 
over twice the length of the pipe; or more generally, 
in which the condensation and rarefaction recur in vir¬ 
tue of any of the modes of vibration of which the column 
of air in the pipe is susceptible. 

§ 110. In general, whenever there is a free communi¬ 
cation opened between the column of air in a pipe and 
the free atmosphere, that point becomes a point of 
maximum excursion of the vibrating molecules, or the 
middle of a ventral segment. At such a point the rare¬ 
faction and condensation assume their smallest possible 
values by the air reducing itself constantly to an equi¬ 
librium of pressure with the external air. Hence, if the 
pipe speak at all, it will take such a mode of vibration 
as to satisfy this condition, but', consistently with this, it 
may divide itself into any number of ventral segments. 
But here there is a practical difference between the 
affections of a vibrating aerial column and those of a 
tense cord. In the case of the cord both ends in prac¬ 
tice must be fixed to secure the requisite elasticity; this 
the air possesses in its natural state, and to make the 
cases analogous we must suppose the cord to be extend¬ 
ed in one direction to infinity, so that its pulses like 
those of the aerial column may run off indefinitely 
never to return. 

§ 111. In cords with fixed extremities all the ventral 
segments must of necessity be complete, no half segment 
can exist. In pipes it is otherwise. The air in a pipe 
closed at one end vibrates as a 
half, not as a whole of such a 
segment. It is owing to this 
that a pipe open at both ends 
can, if properly excited, yield a 


rig. 70. 





ELEMENTS OF ACOUSTICS. 


121 


musical sound. The column of air vibrates in the mode 
represented in the figure, in which there is a node in 
the middle, and each ventral segment is only half a 
complete one. 


Node in the 
middle of the 
entire segment 


§112. To find the time of vibration or the number Pipe open at 
of vibrations in a given time r> both ends; 

corresponding to any mode of ----- 

vibration, denote by m the T * > < 

number of nodes in a pipe 

open at both ends; the number of complete ventral 
segments between them will be 


Fig. 71. 


m — 1; 


Number of 
complete ventral 
segments; 


and denoting the length of the pipe in feet by Z /; , the 
length of each complete ventral segment will be 



Length of each 
segment; 


and denoting the yelocity of sound by V, and the time 
required for the sonorous pulse to traverse one seg¬ 
ment by T, we shall have 


T= 


1. Li. 

m V 


Time of 

(43) describing < 
J segment; 


and this is the time of vibration of the middle section 
of the segment to which the sound corresponds. 

The number of vibrations per second being iT, there 
will result 





(44) Number of 

vibrations in a 
second; 


and the pitches of the series of tones which the pipe 
can be made to deliver will be expressed by the values 












122 


NATURAL P.HILOSOPHY. 


Pitches of the 
tones delivered 
by the pipe. 


Pipe closed at 
one end; 


Number of entire 
segments; 


Length of each; 


Time of 
describing one 
segment; 


Pitch, or number 
of vibrations per 
second; 


Series of pitches. 


of JV, determined by making successively m = 1, m = 2, 
m = 3, &c., or by 


1 V 9 Y q V o 
1. , 2., 3. —, &c. 

-L'u A/ A/ 


§ 113. In the case of a pipe stopped at one end, the 
closed end must be regarded as 
a node; and denoting, as before, 
the number of nodes by m, the 
number of complete ventral seg¬ 
ments will be m — 1, and one 
half segment at the open end, or 


Fig. 72. 


m — 1 + -J- = 


2 m — 
~~~ 2 ~ 


1. 

—5 


and the length of each complete one, in feet, will be, 


2 A, . 

2m — 1 ’ 


and the time T 7 , required for a sonorous pulse to tra¬ 
verse each segment, will be given by 


T = 


2 m — 1 V 


. . (45) 


and the pitch by 


JSr = 





and making m = 1, m = 2, m = 3, &c., the pitches of 
the tones will become 


t v . 3 y , f o 

*■ l: *-t: T - r ; &c - 















ELEMENTS OF ACOUSTICS. 


123 


§ 114. Lastly, in the case of a pipe stopped at both Pipe closed at 
ends, the number of nodes, in- both ends; 

eluding the two ends, being m, Fi s- 7a 

the number of ventral segments 

& . < - > <; - > 

will be m — 1; the length in ---- 

feet of each will be 



Length of each 
segment; 


the time, 


T = _JL_. -ku • 

m — 1 V 9 


and the pitch, 


xY = (m -1) . 



and the series of pitches, 


Time of 

(47) describing on* 
segment, 


Pitch, or number 
(48) of vibrations 
per second; 


1. X. ; 2. H ; 3. — ; &c.(49) ^ " f > ,ltch ”- 

L,, L u L„ 

Taking, therefore, the.number of vibrations performed 
in the fundamental note in one second as unity, the series 
of harmonics will run thus: 


In a pipe stopped at both ends . 

« « u 0 p en a t both ends . . . 

“ u u stopped at one end and ) 
open at the other f 


1, 2, 3, 4, 5, &c. 
1, 2, 3, 4, 5, &c. 

1, 3, 5, 7, 9, &c.; 


Series of 
harmonics. 


it being recalled that, Equations (44), (46), and (48), in 

the last series, the fundamental note is an octave lower These sounds 

than in the other two. produced by 

rr, t , blowing into a 

To produce these sounds by blowing into a pipe, itp ipe; 













124 


NATURAL PHILOSOPHY. 


Fundamental 
tone heard first; 


Biot’s 

experiments; 


Explanation of 
these results. 


The air is the 
Bounding body; 


Verification. 


will only be necessary to begin with as gentle a blast as 
will make the pipe speak, and to augment its force gra¬ 
dually. The fundamental tone will first be heard, which 
will increase in loudness till suddenly it starts up an 
octave; that is, passes the interval between notes whose 
vibrations are as one to two. By adapting an organ- 
bellows to regulate the blast, M. Biot succeeded in draw¬ 
ing from a pipe all the harmonic notes represented 
by the series of natural numbers up to 12, inclusive, 
except 9 and 10; the reason for failing to produce 
these two is not stated. 

The rationale of this continued subdivision of a vibrat¬ 
ing column as the force of the blast increases is obvi¬ 
ous. A quick, sharp current of air is not so easily turned 
aside from its course as a slow one, and when thrown 
into a ripple by any obstacle will undulate more rapidly. 
Consequently, on increasing the force of the blast a pe¬ 
riod will arrive in which the current can7iot be diverted 
from its course and return to it as slowly as required 
for the production of the fundamental note, and the 
next higher harmonic will be excited. 

§ 115. That it is the air which is the sounding body 
and not the material of the pipe, appears from the 
fact that the kind, thickness, or other peculiarities of 
the latter, make no difference in the tone in regard to 
pitch. A pipe of paper, lead, glass, or wood, of the 
same dimensions, gives, under the same circumstances, 
the same pitch. The qualities of the tone are often 
different, but this is owing to the feeble vibrations of 
the molecules of the material of the pipe produced by 
those of the contained air. 

§ 116. Putting the two values of JV, in Equations (41) 
and (46), equal, we find, 

2m - 1 V __ n : V2 g L, 

2 -Z„ 2 £ 


Equation. 





ELEMENTS OF ACOUSTICS. 


125 


whence 


Y — n L ti V‘2gL, 

2m—1 * L 


/ka\ Velocity of sound 
W U ) inagas; 


|and making m and n each unity, 


y — ^9^, 

L 


(51) 


Same reduced; 


which furnishes a ready means of finding the velocity 

of sound in any gas or vapor. For this purpose, fill a 

pipe of known length with the gas in question, and set 

it to vibrating by any proper means, so as to call forth 

its fundamental tone. Adjust the bridges of a Mono- Practical use of 

chord so that the fundamental tone of its string shall 

have to the ear the same pitch; measure the length 

of the string between the bridges and substitute this 

length for L in Equation (51), and the velocity sought 

becomes known. It was by this method that Chladni, 

V" aknees, Fkameyer and Moll ascertained the velocity 
of sound in various media. 

For a detailed account of the structure and manage- Account of, pipe8, 
ment of the embouchures of pipes, and a vast amount 
of interesting matter on the subject of reeds, &c., &c., 
the reader is referred to Sir John Hekschel’s most va¬ 
luable Monograph of Sound, articles 197 to 207, inclu¬ 
sive, as published in Yol. IY. of the Encyclopedia 
Metropolitana. 








126 


NATURAL PHILOSOPHY. 


Vibrations of 
bars; 


Transversal and 
longitudinal; 


Laws governing 
the pitch in 
transversal 
vibrations; 


Means of 
producing these 
vibrations in 
bars. 


Longitudinal 
vibrations in 
bars; 


VIBRATIONS OF ELASTIC BARS. 

§ 117. Bars of a cylindrical or prismatic shape- are 
susceptible of sonorous vibrations as well as cords, and 
columns of air. But as such bodies are nearly equally 
elastic in all directions, transversely as well as longitu¬ 
dinally, their vibrations do not obey the same laws as 
those of strings. Transversal vibrations may be excited 
by striking a bar crosswise, and longitudinal vibrations 
by striking it in the direction of its length. 

In bars made of the same substance, the acuteness 
of the-pitch in transversal vibrations is directly as the 
thickness, and inversely as the square of the length of 
the bar. In bars made of different substances, it is 
foimd that the degree of the body’s elasticity greatly 
influences the character of the pitch; thus steel gives a 
higher pitch than brass. 

To produce these vibrations, the bar may be either 
secured at both ends, or its ends 
may be made merely to rest on 
some fixed objects; or one end 
may be fastened while the other is 
free, or lastly, both ends may 
be free, the rods being support¬ 
ed at two points. 

We have an illustration of 
these kinds of vibrations in the 
jews-harp, musical boxes, &c. 

§ 118. When a bar is struck upon one of its ends in 
the direction of its length, the blow will give rise to a 
condensed pulse, which will proceed towards the other 
end like that of a column of air. It will be reflected 
back and forth alternately at the two ends, according 
to the principles of §106 and §107, and this will con¬ 
tinue till its living force is wholly transmitted to the 


Fig. 74. 












ELEMENTS OF ACOUSTICS. 


127 


air and wasted in space. If the rod be of glass, the solid rods. give 
sound emitted will be extremely acute unless its length 
be very great; much more so than in the case of a columns of air; 
column of air of the same length. The reason of this 
is, the greater velocity with which sound is propagated 
in solids than in air. When the bar is short the re¬ 
flexions at the ends, which determine the successive 
impulses upon the air and therefore the pitch, succeed 
each other with great rapidity. The velocity in cast 
iron, for example, being 10 \ times that in air, a rod Glass > stee1 ’**• 
of this metal will yield a fundamental sound when lon¬ 
gitudinally excited, identical with that of an organ-pipe 
of of its length, stopped at both ends, or 2 Y °f its 
length, open at one end. 

The laws of longitudinal vibrations have nothing in 
common with the transversal, except that the acuteness Laws of 
of the sound emitted varies inversely as the length 0 f longitBdinalaTld 

° transversal 

the bar, the reason of which is obvious. The sounds vibrations differ; 
produced by the longitudinal vibrations are, without ex¬ 
ception, higher than those yielded by the transverse 
vibrations of the same body. They are little if at all 
influenced by the thickness, or, iii the case of wires of 
considerable thickness, by tension. As in the case of 
transversal vibrations, the sounds emitted from bars of 
equal dimensions depend upon the nature of the ma¬ 
terial. 

Longitudinal vibrations may be generated in elastic 
bars, by holding them in the middle between two An¬ 
gers, and rubbing repeatedly one 
of the ends with the Angers of 
the other hand. In experiment¬ 
ing on glass tubes the friction- 
apparatus represented in the 
A "lire will be found convenient: 
a , a, are two pieces of wood hol¬ 
lowed out, having their cavity 
padded with cloth or leather ; 


Longitudina. 
vibrations 
produced; 






128 


NATURAL PHILOSOPHY. 


Friction 
apparatus; 


* 


Nodal points 
found; 


Musical 

instruments. 


c c, is a steel spring connecting 
them, and d, are two rings in¬ 
tended to receive the fingers with 
which the friction is excited. 

Moisten the padding with spirit 
of wine, and sprinkle on it a little 
finely pulverised pumice-stone. 

If metal or wooden bars are 
used, the readiest mode will be 
for the operator to put on a leather glove, on the thumb 
and index finger of which is some pounded resin, and 
with these to rub the rods. 

The existence of nodal points may be verified by sliding 
small paper rings loosely on the rod. 

These vibratory movements have been applied to musi¬ 
cal purposes in some instruments. Kaufmann’s Harmo- 
nichord and Chladni’s Eujphon act on this principle. 


Fig. 75. 



vibrations § 119. Beside the two species of vibration described al- 

producedinbars rea( jy elastic rods admit* of a third, viz., that bv rotation. 

It is most easily generated in cylindrical bodies, by secur¬ 
ing one end in a vice, and communicating to the other a 
rotatory motion by means of a bow or by friction. An 
alternate expansion and contraction ensue in a direction 
perpendicular to its axis. Different high and low notes 
succeed each other, of which, as yet, no use has been 
made in music. 

\ 


OF THE VIBRATIONS OF ELASTIC PLATES AND BELLS. 

vibrations If elastic plates, of glass or metal in particular, be held 

JET m tightly either by the fingers or by means of a clamp, at 

any one point, and the bow of a violin be drawn across 
the edge of the plate, sonorous undulations are imme¬ 
diately produced. 

These oscillations resemble those of elastic rods, inas- 





ELEMENTS OF ACOUSTICS. 


129 


much as the surface is divided into a greater or less num¬ 
ber of perfectly symmetrical parts, and such as are con¬ 
terminous, vibrate in opposite directions. 

The boundary lines of these several parts are all in a chiadnrs 
state of repose, and form nodal lines ; their position de- 80norou3 fl " ure3; 
pends on the places at which the plate is held and ex¬ 
cited, as one of these nodal lines invariably runs through 
the point at which the plate is held, whilst the plate itself 
receives the vibratory motion at the other point. These 
lines form certain peculiar figures, called, after their dis¬ 
coverer, Ciiladni’s Sonorous Figures . 

To make these figures visible, and to render them per- Meansof making 
manent, strew some light sand or dust over the plate; these visible; 
they may also be seen if a small quantity of water be 
poured on the plate, nay, even by the rays of light 
falling on it. Wheatstone remarks that, in using the 
last-named mode, still more delicate divisions in the 
figures were observable. 

These sonorous figures are composed sometimes of Their shapes 
right lines, sometimes of curves either parallel to or ^sTo/thT 
intersecting each other. The shape of the plate greatly P late ; 
affects them, as they are differently arranged, accord¬ 
ing as it may be a square, a rectangle, a triangle, a 
circle, an ellipse, or some other figure. A perfectly dis¬ 
tinct and well-defined figure is produced only when the 
plate gives a very clear sound. 

By experiments made on such plates the following Laws* 
laws were detected by Chladni : 

1. Any particular pitch will always produce the same 
figure with the same plate; but a small change may 
often be produced in the figure by slightly changing 
the place at which the plate is held without causing 

any difference in the pitch. If the pitch be changed, First law; 
the existing figure disappears at once, and a new one 
arranges itself. 

2. The gravest pitch any plate gives is accompanied 

by the simplest figure, and the higher the pitch the more second law; 
complex the figure, i. e. the more nodal lines there will be. 

9 



130 


Third law. 


Experimental 
illustration; 


Union of several 
plates of equal 
size; 


The effects. 


NATURAL PHILOSOPHY. 


3. If similar plates of various sizes be treated in the 
same manner, similar figures will be generated in each; 
by the same treatment, we mean that they shall be 
held at the same point, and that the bow shall pass 
over corresponding points in each. The pitches will, 
however, differ, for the larger plate will give out the 
graver sound; and if their dimensions be equal, the 
stronger will give the acuter pitch. 

§ 120. If the plates be strewed with fine sand, and 
held at the point whilst the bow be made to pass 
at 2>, the figures here depicted will in each case be 
produced. 

Fig. 76. 



A striking effect is obtained by making the same 
figure on several plates of equal size and similar form, 
and then so arranging them as to make one figure on 
a larger scale. The figure thus produced will be both a 
compound and connected one, and such as may not 
unfrequently be met with on a large plate. 

If a large square be formed out of four squares, bear¬ 
ing the figures I. and II., we shall have the following: 


Fig. 77. Fig. 78. 























ELEMENTS OF ACOUSTICS. 


131 


If the large square plates be held at a, touched at a\ Particular case; 
and a bow be drawn across at 5, similar compound fig¬ 
ures will be generated. 

Cymbals , the Chinese Tam-tam or Gong, &c., are prac- Examples; 
tical applications of sonorous plates. 

§121. The vibrating motions of sonorous bells resem- sonorous belts; 
ble those of circular plates. In this case, too, the 
most acute pitch is accompanied by the most complex 
figure. 

To render these vibrations visible, fill a bell-shaped 
glass rather more than half full of water; draw a vio¬ 
lin bow across the rim, and at the same time touch the 
glass at two opposite points of the rim with the fingers. 

The surface of the water will acquire an undulatory mo- Means of making 

tion, and to make the sonorous figures permanent, strew 

the surface of the water with any light and exceedingly 

fine powder, as semen lycojpodii. If the point excited 

by the bow be at a distance of 45° from that touched 

by the finger, a four rayed star marked III. of the last 

article will result; but if the distance be 30°, 60°, or 

90°, the six-rayed star, marked IV., will appear. 

Such a cup gives musical sounds when rubbed with the They g{ve 
moistened finger. The vibrations of the glass in this musical sounds, 
case result from torsion, and this is the principle of the 
well known finger glass. 


COMMUNICATION OF VIBRATIONS. 

§ 122. The numerous experiments of M. Savabt abun- communication 
dantly show that the molecular motions of one body are of molecular 
communicated to another, when there exist between them 
any intervening media, and this the more effectually as 
the connection is the more perfect. But not only this; 
they also show that the molecules of the neighboring 
bodies are agitated by motions both similar in period and 
parallel in direction to those of the original source of mo-its peculiarity; 



.132 


NATURAL PHILOSOPHY. 


Experimsntal 
illustration; 


Effect when the 
glass and 
membrane are 
parallel; 


Effect when 
inclined to one 
another; 


When 

perpendicular to 
one another; 


When shifted 
laterally; 


When the plate is 
revolved about 
its vertical 
diameter. 


tion. Of these experiments we have only room for such 
as have a direct bearing upon the nature and structure of 
our organs of hearing. 

§ 123. Take a thin membrane, moistened tissue paper 
will answer every purpose, and stretch it over the mouth 
of a common bowl or finger glass, place it in a horizontal 
position and strew fine sand over its surface. Hold a glass 
plate, covered with fine sand or dust, horizontally and di¬ 
rectly over the membrane, and set it in vibration so as to 
form Chladni’s acoustic figures; these figures will be im¬ 
mediately and exactly imitated in the sand on the mem¬ 
brane, and this will be the case to whatever lateral posi¬ 
tion within the sphere of sufficient action to move the par¬ 
ticles of sand, the plate may be shifted, provided it retain 
its parallelism to the membrane. 

§ 124. But instead of shifting the plate laterally, let its 
}3lane be inclined to the horizon. The figures on the 
membrane will change though 'the vibrations of the plate 
remain unaltered, and the change will be greater, the 
greater the inclination of the plane of the plate. And 
when it becomes perpendicular to the horizon and there¬ 
fore to the surface of the membrane, the figures on the 
latter will be transformed into a system of straight lines 
parallel to the common intersection of the two planes; 
and the particles of sand, instead of dancing up and down, 
will creep in opposite directions to meet on these lines. One 
of these, lines always passes through the centre, and the 
whole system is analogous to what would be produced 
by attaching a cord to the centre of the plate, and, 
having stretched it very obliquely, setting it in vibration 
by a bow drawn parallel to the surface. In a word, the 
vibrations of the membrane are now parallel to its sur¬ 
face, and they preserve this character unchanged, how¬ 
ever the plate be shifted laterally, provided its plane 
be kept vertical. If the plate be made to revolve about 
its vertical diameter, the nodal lines on the membrane 
will rotate, following exactly the motions of the plate. 



ELEMENTS OF ACOUSTICS. 


133 


§125. Nothing can be more decisive or instructive inference; 
than this experiment. It shows us that the motions of 
the aerial molecules in every part of the spherical wave 
propagated from a vibrating centre, instead of diverging 
like radii in all directions, so as to be always perpen- Principle of 
dicular to the wave surface, may be parallel to each*™”®™ 1 ^ 1 
other and to the wave surface. The same holds good 
in liquids also. 


§ 126. So long as the sound of the plate, its mode of 
vibration, its inclination to the plane of the membrane, 
and the tension of the membrane continue unchanged, 
the nodal figure on the membrane will continue the Conditions to 
same; but if either of these be varied, the membrane msure 
will not* cease to vibrate, but the figure on it will be figure; 
changed accordingly. Let us consider separately the 
effects of these changes. 


§ 127. All other things remaining the same, let the 
pitch of the sounding plate be altered, either by loading Difference 
it or changing its size. The membrane will still vibrate, betw ® cn a 
differing in this respect from a rigid lamina , which can rigid lamina; 
only vibrate by sympathy with sounds corresponding to 
its own subdivisions. The membrane will vibrate in sym¬ 
pathy with any sound, but every particular sound will 
be accompanied by its own particular nodal figure, and 
as the pitch varies, the figure will vary. Thus, if a slow 
air be played on a flute near the membrane, each note illustration, 
will call up its particular form, which the next will efface 
to establish its own. 

§ 128. Next suppose the figure of the plate so to vary Change of figure 
as to change its nodal figures; those on the membrane of the plate; 
will also vary ; and if the same note be produced by dif¬ 
ferent subdivisions of different sized plates, the nodal 
figures on the membrane will also be different. 



134 


NATURAL PHILOSOPHY. 


Effect of change 
of tension; 


Effect of 
moisture 


Secondary nodal 
lines; 

Inference; 


Explanation; 


Sensibility of 
some 

membranes; 


Exploring 

membranes 


§ 129. If the tension of the membrane be varied ever 
so little, material changes will take place in* its nodal 
figures. Hygrometric variations are sufficient to produce 
these changes. Indeed, the fluctuations arising from this 
cause were so troublesome in the case of tissue paper, 
that it became necessary to coat the upper surface with a 
thin film of varnish. By far the best substance for ex¬ 
hibiting the results of these beautiful experiments is var¬ 
nished paper. Moisture diminishes the cohesion of 
the fibres, and renders them nearly independent of each 
other, and sensible alike to all impulses. 


§ 130. Between the nodal lines formed by the coarser 
particles of sand, others are occasionally observed, formed 
only of the finest dust of microscopic dimensions. This 
is a most important fact, as it goes to show that diffe¬ 
rent and higher modes of subdivision coexist with the more 
elementary divisions which produce the principal figures. 
The more minute particles are proportionally more re¬ 
sisted by the air than the coarser ones, and are thus 
prevented from making those great leaps which throw 
the coarser ones into their nodal arrangement. They 
rise and fall with the greater divisions of the surface, 
and are only affected by those minute waves which 
have a smaller amplitude of excursion and occur 
more frequently, and form their figures as though the 
others did not exist. These secondary figures often ap¬ 
pear as concentric rings between the primary ones, and 
frequently the centre of the whole system is occupied 
as a nodal point. 


§ 131. So sensitive are some varieties of stretched mem¬ 
brane to the influence of molecular motion that they 
have been employed with success in detecting the ex¬ 
istence and exploring the extent and limits of the most 
delicate, continuous and oppositely vibrating portions of 
air. When so employed they are called exploring mem - 



ELEMENTS OF ACOUSTICS. 


135 


branes. The most highly interesting application of the Application of 
properties of stretched membrane is in the “ membrana the properties of 
tympani” of the ear. . “e; 



THE EAR. 

§132. The auditory apparatus, called the ear, is a Essential part8 
collection of canals, chambers, and tense membranes, of the ear; 
whose office is to collect and convey to the seat of 
hearing, the vibrations impressed upon the air by sono¬ 
rous bodies. 

Beginning on the exterior and proceeding inwards, 










13 G 


NATURAL PHILOSOPHY. 


"Wing; 

Auditory duct; 


Cavity of the 
drum; 


Labyrinth; 


Fig. 80. 



Vestibule; 
Fenestra ovalis 
and conclilea; 


we find a cartilaginous funnel A A , called the wing / 
a canal b b , called the 
auditory duct , leading to 
an interior chamber i?, 
called the cavity of the 
drum ; and behind this a 
system of canals of con¬ 
siderable complexity, call¬ 
ed the labyrinth , consist¬ 
ing of three semi-circular 
tubular arches m , m, m, 
originating and terminat¬ 
ing in a common hall ?i, 

called the vestibule , which communicates with the cavity 
of the drum by a small opening l, called the fenestra ovalis , 
and is prolonged in the opposite direction into a spiral 
cavity o, called the cochlea. The auditory duct is closed 
at its junction with the cavity of the drum by a tense 
Dmmof the ear; membrane r, called the drum of the ear, as is also the 
fenestra ovalis by a similar membrane. The whole ca¬ 
vity of the labyrinth is filled with a liquid in which are 
Auditory nerve; immersed the branches of the auditory nerve , wherein 
is supposed to reside the immediate seat of the first im¬ 
pression of sound. Within the cavity of the drum are 
four small bones united by articulations so as to form a 
continuous chain; the first f is called the hammer , tjie 
second <7, the anvil, the third i, the ball , (os orbicularis), 
and the fourth Jc, the stirrup , from the resemblance which 
its shape bears to that of the common stirrup. The han¬ 
dle of the hammer is attached to the drum, and the 
stirrup to the membrane which closes the fenestra ova¬ 
lis ; and thus the aerial vibrations, first collected by the 
funnel-shaped wing of the ear, and transmitted through 
the auditory duct to the drum, are conducted onwards 
by the articulated bones to the auditory nerve in the 
labyrinth, which receives them at the window of the 
vestibule. The cavity of the drum is connected with that 
Eustachian tube ; of the mouth by a canal d, called the Eustachian tube , 


Hammer, anvil, 
ball and stirrup; 




ELEMENTS OF ACOUSTICS. 


137 


which serves to keep the cavity of the drum filled with i* use- 
air of uniform density and temperature; a condition which 
appears to be necessary in order that the different parts 
may perform their functions with accuracy. If this be 
stopped, deafness is said to ensue, but as Dr. Wollaston 
has shown, only to sounds within certain limits of pitch. 

If the membrane which closes the labyrinth be pierced Deafties3 
and its fluid let out, complete and irremediable deaf-proceed, 
ness ensues. From some experiments of M. Floukens 
on the ears of birds, it appears that the nerves en¬ 
closed in the several arched canals of the labyrinth have 
other uses besides serving as organs of hearing, and are other uses of th« 
instrumental, in some mysterious w*ay, in giving animals nerves of the 
the faculty of balancing themselves on their feet and 
directing their motions. 


MUSIC, CHORDS, INTERVALS, HARMONY, ><* 
SCALE AND TEMPERAMENT. 


§ 133. Our impression of the pitch of a musical sound th ™ affect oar 
depends, as we have seen, entirely upon the number impression of a 
of its vibrations in a given time. Two sounds whose muslcalsound ’ 
vibrations are performed with equal rapidity, whatever 
be their difference in intensity and quality, affect us 
with the sentiment of accordance, which we call uni¬ 
son , and impress us with the idea that they are simi¬ 
lar. This we express by saying that their pitch is the 
same, or that they are the same note. The impulses Effectoftwo 

^ sounds in unison; 

wdiich they send to the ear through the medium of 
the air, occurring with equal frequency, blend and form 
a compound impulse, differing in quality and intensity 
from either of its components, but not in the frequency of 
its recurrence, and we judge of it as of a single note 
of intermediate quality only. of two notin 

But when two notes not in unison are sounded to- unl8on: 

10 



138 


NATURAL PHILOSOPHY. 


Concord oi getlier, most persons distinctly perceive botji, and can 

discord. separate them in idea, and attend to one without the 

other. But besides this, the mind receives an impres¬ 
sion from them jointly which it does not receive from 
either when sounded singly even in close succession ; an 
impression of concord or of discord , as the case may be, 
and hence the mind is pleased with some combinations, 
displeased with others, and it even regards many as harsh 
and grating. 


Harmony, chord, 
melody. 


Music. 


Concordant 
sounds; 


Discordant 
sounds; 


Limit of 
simplicity in 
music. 


§ 134. The union of simultaneous and concordant 
sounds, is called Harmony. Every group of simultaneous 
and concordant sounds, is called a Chord in harmony. A 
succession of single sounds makes Melody. To discover and 
discuss the laws of harmony and melody, are the objects 
of musical science ; to apply these laws to the production 
of certain effects in musical composition, is the object of 
musical art. Science and art,' thus employed, constitute 
that department of knowledge properly called Music. 

Now it is invariably found that the concordant sounds 
are those, and those only, in which the number of vibra¬ 
tions in the same time are in some simple ratio to 
each other, as 1 to 2, 1 to 3, 1 to 4, 2 to 3, &c., and that 
the concord is more pleasing the lower the terms of the 
ratio are and the less they differ from each other. 
While, on the other hand, such notes as arise from vibra¬ 
tions which bear no simple ratio to each other, as 8 to 15, 
for instance, produce, when sounded together, a sense of 
discord, and are unpleasant. By the constitution of the 
ear, ratios in which 7 and the higher primes occur are not 
agreeable; why, cannot be told, but simplicity must end 
somewhere, and in music this seems to be about the point. 
This is the natural foundation of all harmony. 


§ 135. The relative effect of any two sounds is found 
to be always the same as that of any other two in which 
the ratio of the vibrations is the same. Thus sounds of 



ELEMENTS OF ACOUSTICS. 


139 


which the vibrations are respectively 12 and 18, produce Compound 
the same effect as those whose vibrations are 40 and 60, 

for same effect; 

18___ 60__ 3_. 

12 ~ 40 ~ 2 ’ 

and we say that according as the first and second sounded 
together, are pleasant or unpleasant, so are the third and 
fourth; also, if an air beginning on the first sound re¬ 
quire an immediate transition to the second, then, the 
same air beginning on the third will require an immediate 
transition to the fourth. 

§ 136. The relative pitch of two sounds is called an | nterva i ; 
interval. Its numerical value is expressed in terms of 
the graver sound, represented by the number of its vibra¬ 
tions in a given time, taken as unity. The value of It8mimericaI 
an interval is, therefore, always found by dividing the value found ; 
number of vibrations of the acuter note in a given time 
by the number of vibrations of the graver note in the 
same time; thus, the interval of two sounds, one of which 
is produced by two and the other by three vibrations 
in the same time, has for its measure f. If 18, 

23, and 30, be the numbers of vibrations of three sounds Examples; 
in the same time, and we wish to find a fourth sound 
which shall be as much above the third as the second 
is above the first, we say, 

18 : 23 :: 30 : * = 30 ' — = 38i. 

18 3 

§ 137. Next to unison , wherein the vibrations of the two 
sounds are in the ratio of 1 to 1, the most satisfactory vibrations in the 
concord is that in which the vibrations are in the ratio 
of 1 to 2. The effect of this is not only pleasing, but 
it always gives rise to the idea of sameness; insomuch 
that if two instruments were made to play together in 
such manner that the sounds of the one should always 




140 


NATURAL* PHILOSOPHY. 


Give the be of twice as many vibrations as the simultaneous sounds 
impressiencf 0 ^] ier they would be universally admitted to be 

different shades ' J J 

of the same air; playing the same air, with only that sort of difference 
which is heard when a man and a boy sing the same 
tune together. 

Two tense string ^ ow j lake a tense string, and call the sound emitted 
whose vibrations from it (7, and, for the reasons given above, let the sound 
are m the ratio °f £ r °m a s t r i n g 0 f double the number of vibrations be 
called C r . Let us seek for the simplest fractions which 
lie between 1 and 2, up to the prime 7, and we shall 
find, 


3455 6 — 36 

25 35 35 45 4 25 55 


series of and these, arranged in the order of magnitude, give, 
wiii produce alter placing 1 and 2 on the extremes, 

agreeable 

musical sounds. 1, j, f, f, |, f, 2. 

A set of tense strings, or of pipes, so arranged that 
the first makes one vibration while the second makes f 
of a vibration, the third f- of a vibration, and so on to 
the last, which makes 2, will emit sounds every one of 
which will be agreeable when sounded with the first. 


Sounds which § 138. But it is found that the frequent repetition of 
are pleasing and g 0un ds which are very near to each other is not pleasing to 

those which are 17 , x . . & 

not so. an uncultivated ear, and that the frequent repetition o* 

sounds too far from each other is not pleasing to the 
ear after a little cultivation. 

Taking the intervals of the above series, we find 
that from the 


1 st to 2 d is | - 7 - 1 = |; 

2d to 3d is | -7- | = §f ; 

3d to 4th is f -f- f = ||; 

4th to 5th is | -T- | = |; 

5th to 6th is f -r | = y; 

6th to 7th is 2 ~ f = |; 


Examination of 
the above series; 




ELEMENTS OF ACOUSTICS. 


141 


The interval between the 1st and 2d, and that between Defects in this 
the 6th and 7th are too great, while the interval between 801 iesofsounds; 
the 2d and 3d is too small for frequent repetition. A 
new sound must, therefore, be substituted for the second 
one of the scale, and of such value as to increase the 
^interval between the second and third and diminish that Additions 
between the first and second, while an additional sound ieqmred ' 
must be interpolated between the 6th and 7th. Denote 
the first of these by x and the second by y, then will the 
series of ratios stand, 


1 ™ 5 4 3 5 . 9 . 

- L > 45 ¥5 ¥5 35 #5 " 5 


Form of an 
improved series; 


making seven in all, for the octave is but the same 
note with a different pitch. 

But upon what principle shall the values of these 
new sounds be determined, seeing we cannot have any Principles by 

. 1 .. _ which the series 

more simple consonances with the fundamental sound is improvcd . 
whose vibrations are represented by 1 ? The answer is, 
we must take those sounds which make the simplest con¬ 
sonances while they give with the remaining sounds the 
greatest number of consonances. 

The consonance indicated by the interval from the 


4th to 8th is 2 a = 

5th to 8th is 2 § = 

4th to 6th is f -r | = 


1 • 

2 5 Consonances; 

4 . 

3 5 
5 _ 

4 * 


Now, let the sound a?, have between it and the 5th an First 
interval equal to that between the 5th and the 8th; then suppositlon; 
will 

3 rrt - 4 

¥ • X ~ ¥5 

whence 


x = 


Consequence; 


and the first three sounds of the series will stand 1, f, £, fo^ h ; ree 




142 


NATURAL PHILOSOPHY. 


Intervals. 

giving intervals f and y, already found in another part 
of the series. 

Second 
supposition; 

Again, let the interval between the 5th and 7th be 
equal to that between the 4th and 6 th, and we shall 
have 

n § —I— 3. JL 

y • 2 45 

and 

Consequence; 

II 


I^st three sounds tllus mak ing the last three sounds f, V and 2 , and giv- 
and their ing the consecutive intervals of f- and }£, both of which 


intervals. 

o • * 

are found in another part of the series. Replacing a? 
and y by their values, we have 

Diatonic scale; 

0 , D , E , F, G , A , B , O' 

1 , f , f , f , 1 , t , V i 2 » or multiplying by 24 

24,27 , 30,32 , 36,40 , 45,48, 

1st, 2d , 3d , 4tli, 5th, 6th, 7th, 8th. 

Its effect 
universally 
agreeable; 

This is called the natural, or diatonic scale. When 
all its sounds are made to follow each other in order, 
either upwards or downwards, the effect is universally 
acknowledged to be pleasing, and all civilized nations 
have agreed in adopting it as the foundation of their 
music. 

Note, its name, 
and place 
indicated; 

Each sound in the scale is called a note , and takes the 
name of the letter immediately above it ; and its place 
in the order of acuteness from the fundamental note is 

Illustrations; 

expressed by the ordinal number below it. Thus, count¬ 
ing the vibrations of the fundamental note unity, the 
note whose vibrations are f, is named E\ and it is a third 
above <?, regarded as the fundamental note ; in like man¬ 
ner the note whose vibrations are f, is A, and it is a 
sixth above C. The 8th from the fundamental note, or 
G\ is called an octave above C. Again, we say A, is a 
fourth above E\ and E : is a fourth below A, as would be 





ELEMENTS OF ACOUSTICS. 


143 


manifest by simply sliding the scale to the right or in- Confirme(15y 
verting it, so as to bring the number 1, under the note reference to the 
of reference regarded as the fundamental note. scala 

§ 139. This diatonic scale, which is obtained from the Essential 
series of sounds affording the simplest concords with the ( i ualities of the 
fundamental note, after one alteration on account of the " ’ 

too great proximity of two concordant notes, and one in¬ 
terpolation on account of the too great distance of two 
others, has both of the essential qualities of repetition 
and variety. Thus, writing CD, for the interval from 
C to D, and using like notation for the others, and writ¬ 
ing the names which have been adopted by musicians 
for the several intervals, we have the following 


TABLE. 


CD—F G — A B . . 

D E=GA .... 
EF=BC .... 
CE=EA=GB . 

E G — AC' .... 

DF . 

CF=D G = EA=GC' 

FB . 

C G — EB=FC r . . 


= f ; major tone. 

= V j minor tone, f f of a major. 
= 4|; diatonic semitone. 

= f ; major third. 

= |; minor third. 

= If; |f of minor third. 

= 4; fourth. 

— 4f; flattened fifth.* 

= 4 ; fifth. 


Table of intervals 
formed from the 
diatonic scale. 


DA 


— 4 0 . 
2 7 5 


|f of a fifth. 


C A — D B . . . . =f; sixth. 

EC .= |; minor sixth. 

C B .= y ; seventh .* 

DC .= V ; flattened seven th.f 


CC' 


= 2 • octave. 


We observe here three different intervals between con¬ 
secutive notes, viz.: first, that from C to D = E to G 


* An inharmonious interval when the notes are sour ded together 
t Decidedly more harmonious than the seventh. 









144 


NATURAL PHILOSOPHY. 


Major tone; —A to B — |, and called a major tone ; second, that from 
D toE= G to A — y, called a minor tone ; and 
Minor tone; third, that from E to F — B to C' = ||, called, 

though improperly, a diatonic semitone , being in fact 
Diatonic much greater than half of either a major or minor tone, 
semitone or This interval is also called by some authors a limma. 

limma 


Least number of 
vibrations to 
produce 
continuous 
sound; 


Greatest 
number; 


Peculiarities of 
certain 
individuals; 


Causes which 
render sounds 
inaudible. 


Diatonic scale 
maybe 

continued in both 
directions; 


§ 140. When the vibrations are less numerous than 16 a 
second (M. Savakt says 7 or 8), the ear loses the impres¬ 
sion of continued sound, and in proportion as the vibra¬ 
tions increase in number beyond this, it first perceives a 
fluttering noise, then a quick rattle, then -a succession of 
distinct sounds capable of being counted. On the other 
hand, when the frequency of the vibrations exceeds the 
limit of 24000 a second, all sensation, according to M. 
Savakt, is lost; a shrill squeak or chirp is only heard, and 
according to the observation of Dr. Wollaston, some in¬ 
dividuals, otherwise no way inclined to deafness, are alto¬ 
gether insensible to very acute sounds, while others are 
painfully affected by them. It is probable, however, that 
it is not alone the frequency of the vibrations which ren¬ 
ders shrill sounds inaudible, but also the diminution of 
intensity which, from the nature of sounding bodies, must 
ever accompany a rapid vibration among their elements. 
No doubt if a hundred thousand hard blows per second 
could be regularly struck by a hammer upon an anvil at 
precisely equal intervals, they would be heard as a deaf¬ 
ening shriek; but in natural sounds the impulses lose 
in intensity more than they gain in number, and thus 
the sound grows more and more feeble till it ceases to 
be heard. 

If we add to the diatonic scale on both sides the 
octaves of all its tones, above and below, and again 
the octaves of these} and so on, we may continue it inde¬ 
finitely upwards and downwards. But the considera¬ 
tions above show that we shall soon reach practical limits 
in both directions, growing out of the limited powers of 
the ear. 


Practical limits. 



ELEMENTS OF ACOUSTICS. 


145 


§ 141. By the aid of the ascending and descending Pioce9 of mus, ° 
series of sounds thus obtained, pieces of music which 
are perfectly pleasing may be played, and they are 
said to be in the hey of that note which is taken as the Key ; 
fundamental, sometimes called the tonic note of the Tonic note; 
scale, and of which the vibrations are represented by 1. 

And if such pieces be analyzed they will be found to 
consist chiefly if not entirely of triple or quadruple 
combinations of several simultaneous sounds called chords , chords; 
such as the following: 


C,D,E, F,G,A,B ,C' ,D' ,E' ,F' ,G' ,A' ,B' ,C" 

1 £1 4 3 515 0 1_8 1 0 8 6 1 0 3 0 4 . 
X 5 85453 525 35 8 5 "" 5 8 5 453 5 2535 8 5 ^* 

1st, 2d , 3d , 4th, 5th, 6th , Tth ,8th , 9th , 10th, 11th, 12th , 18th, 14th, 15th. 


1 st. The common or fundamental chord, called also chord of ti» 
the chord of the tonic, which consists of the 1st, 3d and tonic; 

5th ; or the 3d, 5th and octave. This is the most harmo¬ 
nious and satisfactory chord in music, and when sounded 
the ear is satisfied and requires nothing further. It is, 
therefore, more frequently heard than any other, and its 
continued recurrence in a piece of music determines the 
key in which the piece is played. 

2 d. The chord of the dominant. The fifth of the key chord of ti.® 
note is called the dominant, by reason of its oft recur- dominant; 
ring importance in harmonic combinations of a given key. 

The chord of the dominant i& constructed like that of 
the tonic, but on the dominant as a fundamental note, 
and consists of the 5th, 7th and 9th, being the 5th and 7th 
of one scale, and the 2d on the next following scale of 
octaves; or, replacing the latter note by its octave be¬ 
low, the notes of this chord will be 2d, 5th and 7th. 

3d. The chord of the sub-dominant: that is, the chord Chord of the 

. sub-dominant. 

constructed upon the 4th note next below the dominant. 

It consists of the 4th, 6th and 8th; or, replacing the lat¬ 
ter note by its octave below, the notes of this chord be¬ 
come the 1st, 4th and 6th. 

10 



146 


NATURAL PHILOSOPHY. 


False close. 


Dissonance of 
the 7th. 


Short pieces of 
music; 


Change of key 
or modulation, 
avoids monotony; 


example for 
illustration; 


4th. The false close , which is the chord of the 6th, 
its notes being 6th, 8th and 10th, or replacing the last 
two notes by their octaves below, 1st, 3d and 6th. The 
term false close arises from this, viz.: A piece of mu¬ 
sic frequently before its termination (which is always 
on the fundamental chord) comes to a momentary close 
on this chord, which pleases only for a short time, and 
requires the strain to be taken up again and closed 
as usual, to give full satisfaction. 

5th. The dissonance of the 1th, or the combination of 
the 2d, 4th, 5th and 7th. It consists of four notes, and 
is the common chord of the dominant with the note 
immediately below it, or the 7th in order above it. 

§ 142. "With these chords and a few others, music may 
be arranged in short pieces so as to possess considera¬ 
ble variety, but long pieces would appear monotonous. 
In the latter the fundamental note would occur so often 
as to appear to pervade the whole composition, and the 
ear would require a change of key to avoid the feel¬ 
ing of tedium which would naturally arise from such a 
cause. This change of Icey is called modulation. But 
the change is not possible without introducing other notes 
than those already enumerated. 

Suppose, for example, it were desirable to change from 
the key of C to that of G. The chord of the tonic 
in the key of C is composed of the notes CE G ; in the 
key of G, of the notes G B D\ giving the intervals, 

C E— GB — £ and CG= GD' = \. 

In the chord of the dominant, in the key of (7, the 
notes are G B B\ giving the intervals, 

GB=i, and GD'= f, 


the same as before. But the chord of the dominant in 
the key of 6% if it could be formed at all from existing 



ELEMENTS OF ACOUSTICS. 


147 


notes, would consist of D\F\A\ giving the intervals Necessity for 


other notes, 
shown; 


D' F' = |f, and Z>' A' = f f, 


which are very different from the intervals of the com¬ 
mon chord to which they ought to be equal; and in 
order that we may be able to make them equal, we 
must have other notes for the purpose. 

Now D\ being the dominant of G, must be the com- whatthenew 
mencement of the interval, and cannot be altered; not ? s must 

1 7 replace; 

new notes must, therefore, be substituted for F r and A r . 

Denote the vibrations of the new notes by x and y ; 
then, passing to the octave below to avoid the com¬ 
mon factor 2, we must have, 



To find the new 
notes; 


whence, substituting the value f for Z>, 


x = | . | and y — 


That is to say, a change from the key of 0 to that 
of (r, requires for the formation of the chord of the 
dominant in the latter key, two new notes, whose vibra¬ 
tions would be represented respectively by the ratios £ 
and | multiplied by the vibrations in the dominant of G, 

Now, as any note may be taken as the key note, and MultipIicityof 
as the dominant changes with the latter, the number new notes must 
of requisite notes would be so numerous as to render beav0lded ’ 
the generality of musical instruments excessively com¬ 
plicated and unmanageable. It becomes necessary, there¬ 
fore, to inquire how the number may be reduced, and 
what are the fewest notes that will answer. 

For this purpose we remark, that if we multiply the 
values of x and y by 24, to reduce them to the same unit 
as that of the scale of whole numbers in § 138, we lind 



148 


NATURAL PHILOSOPHY. 


How this is 
accomplished; 


Place of first 
new note 
determined; 


Sharp, flat; 


Place of second 
new note; 


Temperament. 


Comma in music. 


9 5 

24 x = g- * -j • 24 = 33 J ; 

24 y = §-•§• 24=491; 

In the scale just referred to we find the numbers 32 
and 36, so that the note whose vibrations are x, is almost 
half way between these two notes, and may be in¬ 
terpolated at that place. It will, therefore, stand between 
F and G , and is designated in music either by the sign 
#. sharp, or b, flat , according as it takes the name of 
the first or second of these letters. Thus it is written 
either %F, or ^G. 

With regard to the note whose vibrations are y, and 
of which the value is 40J, it comes so near to the note 
A, whose value in the same scale is 40, that the ear 
can hardly distinguish the difference between them, so 
that the latter may be used fot it; and though a small 
error of one vibration in 80 is introduced in using A as 
the dominant of D , yet it is not fatal to harmony, and it 
is far better to encounter it than to multiply pipes or 
strings to our instruments for its sake. Besides, these 
errors are modified and in a great measure subdued, by 
what is called temperament , of which the foregoing is 
the origin. 

§ 143. The highest note of the perfect chord of the 
dominant of G, is three perfect fifths above C, and the 
note A', which we have adopted in its place, is the oc¬ 
tave of the 6th above G. The vibrations of the first are 
denoted, by (f) 3 = y, and of the second, by 2 . f = y* ; 
and the interval between will be 

2_7 JL. 1_0 — 11 
3 ’ 3 80 * 

This interval of two notes, one of which rises three per¬ 
fect fifths, and the other an octave of the 6th above the 
same origin is called, in music, a comma. 



ELEMENTS OF ACOUSTICS. 


149 


§ 144. Were any other note selected for the fundamen-No two keys 
tai one, similar changes would be required ; and no two ^lef ° 
keys can agree in giving identically the same scale. All, 
however, may be satisfied by the interpolation of a new 
note within each of the intervals of the major and minor 
tones in the scale of article (138), thus, 

#<?) *D) #G) #A) Interpolation 

C, or > , Z>, or V .E^ F\ or > , G, or V A , or V , i?, C ; required; 

bl)) *>E) t>G) M j t>B) 

and the scale thus obtained is called the Chromatic scale; 

scale. 


§145. But what shall be the numerical values of the Numerical values 
interpolated notes? If it were desirable to make the^^ 0 iated 
scale of article (138), which is in the key of (7, (the vi- notes; 
brations of this note being represented by unity,) as per¬ 
fect as possible, at the expense of the others, there would 
be but little difficulty, as the mere bisection of the lar¬ 
ger intervals would possibly answer every practical pur¬ 
pose, and #C — ■ 5_Z>, might be represented by VI • £; 

#D — *>E, by y/ f • |, and so on ; but as in practice no 
such preference is given to this particular key, and as 
variety is purposely studied, we are obliged to depart 
from the pure and perfect diatonic scale ; and to do so Necessity for a 
with the least possible offence to the ear, is the object of tempmment 
a system of temperament.. If the ear required perfect 
concords, there could be no music but a very limited 
and monotonous one. But this is not the case; per¬ 
fect harmony is never heard, and if it were, would be 
appreciated only by the most refined ears; and it is 
this fortunate circumstance which renders musical com- Perfect harmony 
position, in the exquisite and complicated state in which never heard * 
it at present exists, possible. 

§ 146. To ascertain to what extent the ear will bear 
a departure from exact consonance, let us see what takes 



150 


NATURAL PHILOSOPHY 


Extent to which 
tho ear can bear 
a departure from 
exact 

consonance• 


Explanation; 


Beats. 


Example for 
Illustration. 


place when two notes nearly but not quite in unison 
or concord are sounded together. Suppose two equal 
and similar strings to be equally drawn aside from their 
positions of rest and abandoned at the same instant, 
and suppose one to make 100 vibrations while the other 
makes 101, and that both are at the same distance from 
the ear. Their first vibrations will conspire in produc¬ 
ing sound waves of double the force of either singly, 
and the impression on the ear will be double. But on 
the 50th vibration, one will have gained half a vibra¬ 
tion, and the motions of the aerial molecules produced 
by the co-existing waves from both strings will no lon¬ 
ger be in the same but in opposite directions; and this 
being sensibly the case for several vibrations, there will 
be an interference and a moment of silence. As the 
vibrations continue, there will be a further gain, and 
at the 100th this gain will amount to one whole vibra¬ 
tion, when the waves will agaiii conspire, and the sound 
have recovered its maximum intensity. These alternate 
reinforcements and subsidences of sound are called beats. 

Let n , denote the number of vibrations in which one 
string gains or loses one vibration on the other, m the 
number of vibrations per second made by the quicker, 
and if, the interval between the beats, then will 

m : n :: 1*: t 


whence 


n . 

m 


from which it is obvious that the nearer two notes ap¬ 
proach to exact unison, the longer will be the interval 
between the beats. 


Effect of perfect § 147 ' And llere 5t may bo proper to remark upon 
concords; the effect produced in perfect concords and in those 



ELEMENTS OF ACOUSTICS. 


151 


only which, are perfect. If one note make ra vibrations 
while another makes w, it is obvious that if the vibra¬ 
tions begin together, the rath vibration of the one will 
conspire with the nt\\ of the other, and the effect upon 
the ear of these conspiring vibrations will be similar 
to that of' a third set of which each individual vibra¬ 
tion conspires with every rath vibration of the one and 
every n\h vibration of the other of the concordant notes. 
This third set will give rise to a note graver than either 
of the others, and its pitch will, Equation (41), be the 
same as that of a fundamental note of which the con¬ 
cordant notes may be regarded as harmonics. This 
graver note is called the resultant , and those from which 
it arises, components . Let ra = 3 and n = 2, then, see 
scale of article (138), will the concord be a perfect fifth, 
and the resultant note will be an octave below the gra¬ 
ver of the two components. 

What is true of two notes in perfect accordance may be 
shown to be equally true of several, and hence the ex¬ 
planation of this curious fact, viz.: that if several strings 
or pipes be so tuned as to be exactly harmonics of one of 
them, that is, if their vibrations be in the ratios 1, 2, 3, 
4, &c., then, if all or any number of them be sounded 
together, there will be heard but one note, and that the 
fundamental note. For, all being harmonics of the note 1, 
if we combine them two and two we shall find compara¬ 
tively few but what will give resultants which, with the 
individual notes, will be lost in the united effect or re¬ 
sultant of all the component sounds. But to produce this 
effect the strings or pipes must be tuned perfectly to strict 
harmonics. The effect can never take place on the strings 
of a piano-forte, since they are always tempered. 


These effects for 
two concordant 
notes explained 
and illustrated; 


Resultant and 
components. 


Above 

considerations 
applied to several 
concordant notes; 


Explanation of 
effects. 


§ 148. Now, to resume the question of temperament: 

If we count the notes in the chromatic scale of article Temperameit 
(144), we shall find thirteen, and consequently twelve in- * - 
tervals. Hence, if we would have a scale exactly similar 
in all its parts, and which would admit of playing equally 



152 


NATURAL PHILOSOPHY. 


Beale that would well in any key, the question of temperament would re- 
fnauykey'- 7 ^ 11 ^ uce i nsert i n S H geometrical means between 

the extremes 1 and 2, and the scale would stand, 

1 , 2 yl]f , 2 t2 % 2 y3 % . . 2 y ^, 2 . 

iso-iiarmonic The values of the mean terms are readily computed by 

eeai*. logarithms. This scale, which is one of perfectly equal 

intervals, is called the Iso-harmonic scale. 


Examination of 
the chromatic 
acale and table of 
intervals; 


Ascending the 
scale by fifths; 


Ascending by 
major thirds. 


Vaiue of a fifth, 
of a major third, 
and of an octave. 


§ 149. If we examine the chromatic scale and table of 
intervals in article (139), we shall find that the interval 
from E to F\ and from B to G\ are semitones, and that 
in a perfect fifth' there are, therefore, seven, and in an 
octavp twelye semitones. If, then, we reckon upVards 
by fifths, we shall, after twelve steps, come to a note 
in the ascending scale of octaves of the same name as 
that from which we set out. Beginning with C, for 
example, we shall, after the twelfth remove, arrive at 
another C ; or, which amounts to the same thing, if we 
ascend by two-fifths from C and descend an octave, we 
fall upon D ; in like manner, rising by two-fifths from 
D and falling an octave, we fall upon E, and this pro¬ 
cess being sufficiently repeated, we finally reach G\ the 
octave of G. 

Again, from the same scale and table, we see that 
in a major third, that is, from G to E ,J there are four 
semitones, and hence, if we ascend the scale by major 
thirds we shall, after three steps, arrive at the octave 
of the note from which we started. 

The value of a fifth is f, of a major third }, and of 
an octave 2. How, there is no power of f or of £ equal 
to any power of 2, and nence there is no series of 
steps by perfect fifths or major thirds that can lead to 
any one of the octaves of the fundamental note. Were 
the chromatic scale perfect, twelve perfect fifths should 
be equal to seven octaves, and three major thirds to 



ELEMENTS OF ACOUSTICS. 


153 


one octave; but, as just remarked, neither of these can Reckoning 
be true of perfect fifths or major thirds, for (§) 12 =129,74, or 

and 2 7 = 128, giving a difference of nearly one vibra- by fifths; 
tion in every 64; and (f) 3 = l',953, instead of two. So 
that, if we reckon upwards by major thirds, we fall con¬ 
tinually short; if by fifths, we surpass the octave. The 
excess in this latter case is called the wolf , a name sug- The wolf; 
gested, no doubt, by the fact that the thing which bears 
it has been hunted and chased through every part of 
the scale in the vain hope of getting rid of it. In con¬ 
sequence, it has been proposed to diminish all the fifths 
equally, making a fifth, instead of f, to be equal to 2 T % 
and tuning regularly upwards by such fifths, and from 
the notes so tuned, downwards by perfect octaves. This System of equal 

. temperament; 

constitutes what is called the system ot equal temperament. 

In this system the notes must all be represented by identical with the 
the different powers of 2 TJ , and the system itself is iden- ls ° harmonic ’ 
tical with the Iso-harmonic. Theoretically, it is the sim¬ 
plest possible. It has, however, one radical fault; it 
gives all the keys one and the same character. In any its defects, and 
other system of temperament some intervals, though 0 f the remedy, 
the same denomination, must differ by a minute quantity 
from each other, and this difference falling in one part 
of the scale on one key and in a different part on an¬ 
other, gives a peculiar quality to each, and becomes 
a source of pleasing variety. 

Some have supposed that temperament only applies to General 
instruments with keys and fixed notes. This is a mis- appllcatlon °* 

J temperament; 

take. Singers, violin players, and all others who can 
pass through every gradation of tone, must all temper, 
or they could never keep in tune with each other, or with 
themselves. Any one who should keep ascending by 
fifths and descending by thirds or octaves, would soon 
find his fundamental pitch grow sharper and sharper, 
till he could neither sing nor play; and two violin 
players accompanying each other and arriving at the illustration* 
the same note by different intervals, would find a con¬ 
tinued want of agreement. 



154 


NATURAL PHILOSOPHY 


Construction jfa 
table ; 


Table; 


Its accuracy; 


Its use; 


Three intervals 
and their 
notation; 


Enharmonic 
diesis; 


§ 150. If we take tlie logarithms of the fractions which 
express the intervals from the fundamental note to that 
of any other in the diatonic scale, we shall find, after 
multiplying each logarithm by 1000, to avoid fractions, 
taking the product to the nearest whole number, and 
then the successive differences between these, the following 

TABLE. 


From 

Intervals. 

Ra¬ 

tios. 

Logarithms- 

Ap¬ 

prox. 

Differences. 

C to G 
C to JD 
G to E 
GtoE 
G to G 
Gto A 
Gto B 
Gto G f 

0 

major tone 
major third 
minor fourth 
major fifth 
major sixth 
major seventh 
octave 

1 

9 

8 

5 

4 

4 

3 

3_ 

■3 

1 5 

8 

2 

0,00000 

0,05115 

0,09691 

0,12494 

0,17609 

0,22185 

0,27300 

0,30103 

0 

51 

97 

125 

176 

222 

273 

301 

51=.7=maj.tone 

4 6=t = minor tone 
28=0 =limma 
51=2=maj. tone 

4 6=1 = min or tone 
51=T=maj. tone 
28=0 =limma 


The approximate values for the intervals are true to the 
500th of a tone, an interval far too small for the nicest 
ear to distinguish; these values may, therefore, be used 
in all musical calculations when no very high powers of 
them are taken. Since the logarithm of any interval is 
equal to the logarithm of the higher, diminished by that 
of the lower note, the numbers in the column of differ¬ 
ences may be taken to represent the values of the se¬ 
quence intervals, or intervals between the consecutive 
notes expressed in equal parts of a scale of which it takes 
301 parts to measure an octave. 

And we here perceive again the three different kinds 
of intervals referred to in article (139). They are de¬ 
noted in the table above by the characters T, t , and 0, 
their values being respectively 51, 46 and 28, corres¬ 
ponding to the fractions f, y and || of the article just 
cited. These intervals give rise to what is called the 
enharmonic diesis , which is the interval between the 
sharp of one note and the flat of that next above it, 
and enables us to understand the distinction between 










ELEMENTS OF ACOUSTICS. 


155 


flats and sharps; a distinction essential to perfect har¬ 
mony, but which can only be maintained in practice 
in organs and other complicated instruments which ad- its use; 
mit of great variety of keys and pedals, or in stringed 
instruments or in the voice, where all gradations of 
tone may be produced. 

To understand this distinction, suppose in the course To modulate 
of a piece of music it be desirable to modulate 
the key of C to that of F\ its subdominant. To make 
the new scale of F perfect, its intervals should be the 
same and succeed each other in the same order as in 
the original key of G. That is, setting out from F, 
the sequence of intervals should be T t 6 T t T 0, as in 
the table. Now, this sequence does not take place in 
the unaltered scale of <7, when we set out from any 
note but (7, and if we prolong this scale backward to 
F, the notes will stand 



Notes as they 
stand 

erroneously; 


whereas they should stand, 



Notes as they 
should stand; 


The first two intervals are the same in both. The next First two 
two will agree if we flatten the note B , so as to invert j^' aLsi 
the intervals, or make, 


bB - A = 0 = 28; 


To make the 
next two agree; 


and 


C- *>B = T= 51; 








156 


NATURAL PHILOSOPHY. 


Supposition; 


Consequence; 


Interpolation 
necessary to 
render the two 
scales nearly 
perfect in one 
particular case. 


Another case 
supposed; 


Scale as it stands 
in this case; 


Scale as it should 
stand; 


Conclusions. 


giving by addition 

C — A — T + 0 = 5 1 + 28 = 79 = major third. 

The quantity by .which B must be flattened for this 
purpose is obviously 

T - 0 = 51 - 28 = 23; 


and this is the amount by which, in this case , a note 
differs from its flat. As to the remaining three inter¬ 
vals, the difference between T and t being small, amount¬ 
ing only to 5, (which answers to the logarithm of a 
comma ££,) the sequence T t 0 is hardly distinguishable 
from t TO , and if the note D be tempered flat by an 
T~t 

interval = —> or half a comma, this sequence will 

in both cases be the same, and our two scales-of C and 
F will be rendered as perfect as the nature of the case 
will permit by the interpolation of only one new note. 

But, on the other hand, suppose we would modulate 
from C to B. In this case the scale of 0 will stand 


BODE 

I I 

6 j T I t 
whereas it should be 



A 


T 


B' 


B #C *D 
T t 


E 


*F *G *A 



B' 


The intervals from B to F, and from F to B, are the 
only ones that are equal, and to make the others equal 
would require (7, _Z>, F, G and A to be sharpened, and 
consequently the introduction of no less than five new 
notes. But to confine ourselves to the change from A 
to *A we have 
















ELEMENTS OF ACOUSTICS. 


157 


B — A = T = 51 J Particular case 

taken. 

and 

B - #J.= 6 = 28; 

'consequently, by subtraction, 

A ~ #A = 23 = B ~ Result; 

as before determined. But since the whole interval from 
B to A = T = 51, is more than double this interval, 
the flattened note bB, will lie nearer to B , and the Explanation ; 
sharpened note #A nearer to the lower one A than a 
note arbitrarily interpolated half way between A and 
B , (to answer both purposes approximately,) would be, Diesis left in 
and thus a gap or diesis , as it is called, would be left tIlis case; 
between #A and bB. 

The diesis in this case only amounts to T — 2 (T— S) ^ hat ifc amounta 
= 51 — 46 = 5, equal to a comma, or the tenth part 
of a major tone T\ in other cases it would be greater. 

But in all cases the interval between any note and its 
sharp is considered to be equal to that between the 
same note and its flat. 

§151. Taking each note of the diatonic scale as the Each notc ° f th ® 
mndamental or key note m succession, we shall find, takenasthekoy 
by the same mode of comparison, the following sets of note; 
notes in the several scales—the accent at the top of the 
letter denoting one octave above the key note. 

Names of the Keys. 

67, .Z), E\ F\ G , A, B , O', (natural, 0) 

D , E\ *F, G , A, B , #67, D\ (two sharps, B) sets of notes 

E, »F, *G, A, B, *C, *B, E’. (four sharps, E) am8f0 ” 4 

F, G, A, l >B, O, B, E, F’. (one flat, F) 

G , A, B, 67, Z>, E\ *E, G\ (one sharp, G) 

A , i?, #67, Z>, E\ #6r, A', (three sharps, A) 

B , #67, #Z>, E 9 *F, *G, #A, B\ (five sharps, B) 



158 


NATURAL PHILOSOPHY. 


These scales 
defective by 
sharps; 


-In these scales which have the natural notes of the 
diatonic scale for the key, there are but live sharps, 
whereas there should be seven. Where are the other 
two? If we take *F and #6 Y as the key notes, we 
shall find 


Names of the Keys. 

Result of a *F, ##, iff A, B , #(7, #F r . (six sharps, #F) 

*C, *B, *F, *F, *G, »A, *B, *C. (seven sharps, *C ) 


In like manner, constructing a diatonic scale on b B , 
and on each new flat as it is successively introduced, we 
find the following, in which * the accent at the bottom of 
a letter denotes one octave below the key. 


Names of the Keys. 

B,, C, B,bE, F, G, A, bB. (two flats, b B) 

Same for another b E , F, G , X>A , b B , C', B', b E'. (three flats, b E) 

aupposition. , b£ ^ c ^D, b E, F, . G , t>A . (four flats, b^4) 

bD , btf, if 1 , bG, bA, bB, O', b D'. (five flats, b D) 

bG , *>A, bB, t>C, bB', b E', F', bG'. (six flats, bff) 

bo , bB, I >E, bF,bG,bA, bB,bC'. (seven flats, b C) 


several systems § 152. Assuming the principle that the interval 
hivebeen ament between any note and its sharp is to be equal to that 
devised; between the same note and its flat, a variety of systems 
of temperament have been devised for producing the 
best harmony by a system of twenty-one fixed notes, viz: 
the seven notes of the diatonic scale with their seven 
some of the sharps and seven flats. Among the most remarkable sys- 
most remarkable terns may be mentioned those of Huygens, Smith, Young 
and Lagiee, for an account of which the reader is referred 
to the Encyclopoedia Metropolitana, article, Sound. Yol. 
IY., page 797. 


peculiarity of § 153. But the piano-forte, an instrument in almost 
the piano-forte. un j yerga j_ usej an q 0 f the pigh es t interest to all lovers of 
music, admits of only twelve keys from any one note to 
its octave, and a temperament must be devised which 
will accommodate itself to this condition. 




ELEMENTS OF ACOUSTICS. 


159 


We have already spoken of the division of the octave Ar s uments in 
into twelve equal parts, and have seen that this makes temperament 1 ; 
the fifths all too flat, the thirds all too sharp, and gives a 
harmony equally imperfect in all the keys. It is urged 
in favor of equal temperament that all the keys are made 
equally good, and that in no one does the temperament 
amount to a striking defect; also, that in the orchestra 
there is little chance of any uniform temperament if it be 
not this. Against equal temperament it is urged, how¬ 
ever, as before stated, that it takes away all distinctive Against equal 
character from the different keys, and after all, leaves no temp6rament 
one of them perfect. A piano-forte perfectly tuned by 
the system of equal temperament has to some persons a 
certain insipidity which only wears off as the effect of 
this tuning disappears; insomuch that the best phase of illustration by 

° x x _ 1 ' the piano-fort© 

the instrument is exhibited during the period which pre¬ 
cedes its becoming disagreeably out of tune, or, more 
properly, while it is assuming a state of maltonation; for, 
the transition is only a change from equal to unequal 
temperament, in which the several keys begin to exhibit 
variety of character, until maltonation arrives and makes 
the instrument offensive. 

The best jiracticable way of obtaining a given tempera- use of the 
ment, equal or unequal, is by means of the monochord. monocbord; 
The proper lengths of the strings of this instrument, to 
form the required notes, are first calculated, and after¬ 
wards those of the instrument to be tuned are brought into 
unison with them. Ho tuner can get an equal tempera¬ 
ment by trial; so that the question in practice generally General aimln ' 
lies between all sorts of approximations to equal tempera-practice, 
ment, and as many approximations to some other tem¬ 
perament. 

§154. The mode of proceeding by approximation to Themostnsual 

d . 1 _ n i i i order of 

equal temperament is simply to tune all the filths a little proceeding; 
flat; and the following order is the most usual. The first 
letters represent the note already tuned, the second the 
one which is to be tuned from it; a chord in parenthesis 



160 


NATURAL PHILOSOPHY. 


First step, by 
tuning fork; 


Trials indicated 


Explanation of 
method, and of 
results that 
should be 
obtained; 


Bearings. 


Remark on 
unequal 
temperament; 


Smith’s system; 


represents a trial that should be made on notes already 
tuned, to test the success of the operations as far as it has 
gone. The first step is to put C' in tune by the tuning 
fork ; 

G'; G'G; CG; GG,; G,D; DA; A A,; A,E; 
(C.EG); ED; {G EG; DGB); BB,;-B*F; 
{D*FA); *F«F ,; *F, *G: (A,*C E)-, *G#G; 
(. E#GB); C'F; {FAG'); F*A t ; ( *A,DF); 
*A*A; (*A*D; {*DG*A); #D#G,; (*G,C*D). 


All the semitones are written as sharps whether tuned 
from above or below. Since the fifths are all to be a 
little too small in their intervals, the upper notes must 
be flattened when tuned from below, and the lower 
notes sharpened when tuned from above. In the preced¬ 
ing, the octave C C r is completely tuned, and also the 
adjacent interval *F t C. The rest of the instrument is 
tuned by octaves. The thirds should come out a little 
sharper than perfect, as the several trials are made, 
and when this does not happen, some of the preceding 
fifths are not equal. The parts which are first tuned 
by fifths, and from which all the others are tuned by 
octaves, are called bearings. 


§ 155. In unequal temperament, some of the keys are 
kept more free from error than others, both for the 
sake of variety and because keys with five or six sharps 
or flats are comparatively but little used; these latter 
keys are left less perfect, and this is called throwing 
the wolf into these keys. From equal intervals to those 
which produce what has been called maltonation, there 
is abundant room for the advocates of unequal tempera¬ 
ment to select that particular system most congenial to 
the views of each, and, accordingly, many systems have 
been proposed. Of these we shall only mention two, 
viz.: that denominated by Dr. Smith the system of mean 



ELEMENTS OF ACOUSTICS. 


161 


tones, and that which bears the name of its author, Dr. 
Young. 

The system of mean tones supposes the octave divid- s 7 3tem of 
ed into five equal tones, of which we shall denote the 
value of each by a , and two equal limmas, each hav¬ 
ing the value j3, succeeding each other in the order 
a a (3 a a a (3 instead of Tt& Tt Tb, as in the diatonic scale, 
and such that the thirds shall be perfect, and the fifths 
tempered a little flat. These conditions are sufficient 
to determine the values of a and /3, for, 

5 a + 2 (3 = 1 octave = 3 T + 2 t + 24 
2 a = 1 third = T + t 


whence 


T+ t T— t . 

a = — ; ff = 6 + > 


Use of thi3 
system explained 
and illustrated; 


and substituting the values from the table 


51+46 

a — - 

2 


51 - 46 

48,5 ; p = 28 +-j-— 28,125 


and since the interval from the 1st to the 5th of the 
scale is 


3a+P = 2T+t + 6- T —-^\ 

4 

the fifth by this scale is flatter than the perfect fifth 
by the quantity \(T—t\ that is, by a quarter of a com- Results, 
ma. In this system the sharps and flats are inserted by 
bisecting the larger intervals. 

Dr. Young’s first system is as follows, viz.: Tune Young’s am 
downwards from the key note six perfect fifths, then up- system; 
wards from the key note six imperfect fifths, dividing the 
excess of twelve perfect fifths, above seven octaves, 
li 








102 


NATURAL PHILOSOPHY. 


Explanation. 


Scale of tho 
Chines©, 
Hindoos, &©. 


Effect of small 
intervals. 


Principles of 
music applied in 
conversation. 


Minor scales. 


equally among the imperfect fifths, and observing to as¬ 
cend in the first case, and descend in the second, by 
octaves, when necessary, to keep between the key note 
and its octave. 

§ 156. If we take from the diatonic scale the notes F, 
and j5, which rise from those immediately preceding 
them by semitones, there will remain C, _Z>, E, G, A and 
C' for all the sounds of the octave. This is the original 
scale of the Chinese, Hindoos, the Eastern Islands and 
the nations of Northern Europe. It is the scale of the 
Scotch and Irish music, and the Chinese have preserved 
it to the present time. The character of this scale is 
exhibited by playing on the black keys alone of the 
piano-forte. 

§ 157. The effect of making an interval smaller is to give 
the consonance a more plaintive character. It may 
easily be observed, for example, that the intervals of the 
minor third, E 6r, and minor sixth, E C ' on any instru¬ 
ment, have a sad or plaintive effect as compared with the 
major third, GE^ and major sixth, CA. Almost all per¬ 
sons in ordinary conversation are constantly varying the 
tone in which they speak, and making intervals which 
approach to musical correctness, and the effect of sorrow, 
regret, and the like, is to make these intervals minor. 
Any one with a musical ear, noticing the method of say¬ 
ing u I cannot ,” pronounced as a determination of the 
will, and comparing the same uttered as an expression of 
regret for want of ability, will understand what is here 
meant. Why this is so, no one can tell. But the asso¬ 
ciation exists, and resort is had to those modifications of 
the diatonic scale which are known from experience to 
produce the emotions here referred to. The results of 
these modifications, of which there are several, are called 
Minor Scales , in contradistinction to the diatonic, which 
is called the Major Scale. The change from a minor to the 
major scale is one of the most effective of musical resources. 



ELEMENTS OF ACOUSTICS. 


163 


If we return to the fundamental note G and its conso¬ 
nances, viz.: 

Fundamental 
note and its 
consonances; 

and instead of rejecting ^>E as too near to E\ we discard 
this latter note, and finish by inserting D and B of the 
diatonic scale, we shall have what is called the common 
ascending minor scale, as follows : 


C *>E E F G A C' 

1 6 113 5 0 . 

X 5 3)453)2)3)^5 


G , D * *>E , F , O , A , B , C Ascending mlno. 

11 i scale; 

1 9 6 4 3 5. 1_5 O 

A 5 85 5)352535 8) 


But it is not easy to recognize this as a minor scale in Not easily 
descent, because, in going from C ' to C, there is no dis- 
tinction between it and the major scale till we come to Ascent; 
*>E, or until the scale has produced its principal effect 
upon the ear. To remedy this, A and B are both lowered 
a semitone ; that is, A is made t>A, and B is made t>B, 
thus making t>A a fourth to 1>E, and t>B a fifth to ^E, 
and giving 


C,D, *E, F, G, M, *B , C' 

19 643. 8 90. 

x 5 7F 5 553)25 55 5 5 " 5 

which being reversed, is called the common mode of Descending th» 
descending the minor scale. minor scaia 

Again, if we retain B of the major scale and lower 
A, we have 

C,D, t>E, F, G, M, B , C' 

19 .6 4 3 JB 1_5 Q 
A 5 8 5 5 5 If 5 2 5 5)8) ^5 

which is a mild and pleasing scale both in ascent and gcbnei(Jer , s 
descent, notwithstanding the wide interval between A principal minor 
and B. Its harmonics are more easy and natural than 8cale- 
the other, and Schneider makes it, in his Elements of 



164 


NATURAL PHILOSOPHY. 


Harmony, a principal minor scale, and treats all others 
as incidental deviations. 


Any system of §158. ¥e shall now show how we may, from the 
m^Mbe examined theory of the scale, examine any system of tempera¬ 
ture scale; ment; and as the method will be rendered the more 
obvious by applying it to a particular example, we shall 
take the system of Dr. Young just described. 

Let all the intervals be expressed in mean semitones , 
as the unit. There being twelve semitones in the oc¬ 
tave, we have one semitone equal to the logarithm of 
2 divided by 12, or 


0,30103 

12 


= 0,0250858 ; 


Method a nd dividing the logarithm of the major tone = f, that 
•of the minor tone = y, that of the diatonic semitone 
= If, and the excess of twelve perfect fifths over seven 
octaves = 0,00588 by this value of the mean semitone, 
we shall find 

System of d» 1 major tone = 2,039100 mean semitones, 

Young taken; 1 minor tone = 1,824037 u “ 

1 diatonic semitone = 1,117313 “ w 

Excess of 12 fifths over 7 octaves = 0,234600 " 


In tuning upwards, each fifth is to be flattened by 
one-sixth of 0,234600, or by 0,039100. In the equal tem¬ 
perament the wolf is replaced by twelve equal whelps; 
here by six, but of double the size. 

How, a perfect fifth is composed of 

Example for 2 major tones = 4,078200 

illustration; 1 minor tone = 1,824037 

1 diatonic semitone = 1,117313 

Perfect fifth == 7,019550 

Deduct . . . 0,039100 


Flattened fifth 


= 6,980450 







ELEMENTS OF ACOUSTICS. 


165 


Then taking C for the key note, 


C' . . 

12,00000 

G . . 

0,00000 

- 5 th 

. 7,01955 

+5“ . . 

6,98045 

F . . 

4,98045 . (1) 

G . . 

6,98045 . (1) 

+8“ 

12 

+5 . . 

6,98045 

F' . . 

16,98045 

&’ . . 

13,96090 

-5“ 

. 7,01955 

—8** . . 

12 

*A . . 

9,96090 . (2) 

D . . 

1,96090 . (2) 

-5“ 

. 7,01955 

+5 th . . 

6,98045 

*D . . 

2,94135 . (3) 

A . . 

8,94135 . (3) 

+8'* 

12 

+5 tA . . 

6,98045 

#&' . . 

14,94135 

E’ . . 

15,92180 

— 5 th 

. 7,01955 

-S th . . 

12 

*G . . 

7,92180 . (4) 

E . . 

3,92180 . (4) 

— 5 th 

. 7,01855 

+5^ . . 

6,98045 

. . 

0,90225 . (5) 

B . . 

10,90225 . (5) 


12 

+5^ . . 

6,98045 

*C'. . 

12,90225 

#F' . . 

17,88270 

— 5 th 

. 7,01955 

-8 fA . . 

12 

*F 

5,88270 . (6) 

#F . . 

5,88270 . (6) 

Collecting these intervals for all the notes from C to 
C\ we have 

C . 

. . 0,00000 

#F . . . 

5,88270 

*c . 

. . 0,90225 

G . . 

. 6,98045 

D . 

. . 1,96090 

*G. . . 

7,92180 

*D . 

. . 2,94135 

A . . 

. 8,94135 

E . 

. . 3,92180 

*A. . . 

9,96090 

F . 

. . 4,98045 

B . . 

10,90225 


Example 

continued; 


Results collected 


As the most important chord is that of the tonic, we 
form our idea of the effect of each key, from the effect 
of the temperament upon this chord, judging of the 
character of the key by the amount and direction of 





















166 


NATURAL PHILOSOPHY. 


Explanation; the temperament upon the third and fifth, which with 
the key make, as we have seen, the chord in question. 
Now, a major third is composed of 


Value of a major 
third; 


1 major tone = 2,03910 mean semitones, 
1 minor tone = 1,82404 “ “ 

Major third . . 3,86314 “ “ 


A minor third is composed of 

1 major tone = 2,03910 mean semitones, 

1 diatonic semitone =1,11731 “ “ 

Value of a minor Minor third . . . 3,15641 “ 46 

third; - 

and hence the intervals for the chord of the tonic are 


For a major key . . 3,86314 and 7,01955 
“ minor “ . . 3,15641 and 7,01955. 


Conclusions. 


Method of 
examining any 
particular key. 


To examine any particular key, take the numbers from 
the preceding table opposite the notes of the tonic chord, 
adding twelve to make the octave when necessary; sub¬ 
tract the number of the key note from each of the 
other two, and the remainders will give the tempered 
intervals ; from these remainders subtract the correct in¬ 
tervals above, and these second remainders will give the 
amount and direction of the temperament. For exam¬ 
ple, let us examine the key of A ; we find 


Example for 
illustration. 


A . 8,94135; #C' . 12,90225; E’ . 15,92180 
8,94135 8,94135 

Tempered intervals 3,96090 . . . 6,98045 

Perfect intervals . . 3,86314 . . . 7,01955 

Temperaments . . + 0,09776 . . — 0,03910 

whence we see that the first interval is sharper and the 
second flatter than perfect, the sign +, indicating sharper, 
and the sign —, flatter. 

END OF ACOUSTICS. 












ELEMENTS OF OPTICS. 


§1. The principle by whose agency we derive our Light 
sensations of external objects through the sense of sight, 
is called light ; and that branch of Natural Philosophy 
which treats of the nature and properties of light, is 
called Optics. 0pths 


§ 2. There exists throughout space an extremely at- Principle of 
tenuated and highly elastic medium called ether. This 
ether permeates all bodies, and the pulsations or waves 
propagated through it, constitute the principle of light. 

The eye admitting the free passage of the ethereal Sensation of 

J t ° . r . sight produced ; 

waves into it, the sensation of sight arises from 
the motions w T hich these waves communicate to cer¬ 
tain nerves which are spread over a portion of the 
internal surface of that organ; we therefore see by a Analogy between 

0 . % the sensations of 

principle in every respect analogous to that by which sight and sound, 
we hear • the only difference being in the nature of 
the medium employed to impress upon us the motions 
proper to excite these different kinds of sensations. In 
the former case it is the ether agitating the nerves of 
the eye, in the latter, the air communicating its vibra¬ 
tions to the nerves of the ear. 


§ 3. Some bodies, as the sun, stars, &c., possess, in 


Self-luminous 

bodies; 


their ordinary condition, the power of exciting light, 
while many others do not. The first are called self- 
luminous, and the second non-luminous bodies. All 
substances, however, become self-luminous when their .. 
temperature is sufficiently elevated, or when in a state 


Non-luminoM 



168 


NATURAL PHILOSOPHY. 


Insects that 
possoss the 
power of 
exciting light. 


Self-luminous 
bodies visible; 


N on-luminous 
rendered so. 


Medium. 


Waves of light 
spherical in 
homogeneous 
media; 


OeometricA 

illustration. 


Wave front not 
spherical in 
heterogeneous 
media 


of chemical transition ; and some organisms, as the glow¬ 
worm, fire-fly, and the like, are provided with an appa¬ 
ratus capable of exciting ethereal undulations and of 
becoming self-luminous when thrown into a state of 
vibration by these insects. 

Self-luminous bodies are seen in consequence of the 
light proceeding directly from them; whereas, non-lu- 
minous bodies only become visible because of the light 
which they receive from bodies of the self-luminous 
class, and reflect from their surfaces. 

§ 4. Whatever affords a passage to light is called a me¬ 
dium. Glass, water, air, Torricellian vacuum, &c., are media. 

§ 5. Waves of light, like those of sound, proceed from 
any disturbed molecule as a centre, with a constant velocity 
in all directions, through media of homogeneous density. 
The front of the luminous wave in such media is, there¬ 
fore, always on the surface of a sphere whose centre is 
at the place of primitive disturbance, and whose radius 
is equal to the velocity of propagation multiplied into 
the time since the wave began. Thus, if a molecule 
of ether be disturbed at <7, and the 
velocity of propagation be denoted 
by V, and the time elapsed since 
the disturbance by t , then will the 
front of the wave at the expiration 
of this time be upon the surface of 
a sphere whose centre is at G and 
radius C A = V. t. 

If the medium through which the 
wave moves be not homogeneous, the 
shape of the wave front will not be 
vary from that figure in proportion 
parts from perfect homogeneousness. 


Fig. 1. 



spherical, but will 
as the medium de- 


§ 6. The circumstances attending the propagation of 
luminous and sonorous waves are similar. The intensity 



ELEMENTS OF OPTICS. 


169 


of light, like that of sound, depends upon, and is direcLy intensity of 
proportional to the amount of molecular displacement. llght ' 

It is, therefore, Acoustics, § 53, inversely proportional to 
the square of the distance from the original luminous 
source. 

§ 7. ¥e have seen, Acoustics, § 16, that in wave pro-To demonstrate 
pagation through a homogeneous me¬ 
dium, the displacement of a mole¬ 
cule 0, from its place of rest at one 
time, becomes a source of displace¬ 
ment at a subsequent time for an in¬ 
definite number of molecules situat¬ 
ed on the surface of a sphere ME, 
whose centre is at 0 , and of which 
the radius is equal to V. t ; that these 
numerous disturbances become in 
their turn so many sources of disturb¬ 
ance for any single molecule as O', in front of the wave, 
and that the amount of O' ’s displacement from its place of 
rest will be found by compounding the displacements due 
to all these sources, after estimating the amount due to 
each separately. 

To ascertain the effect of this process of composition, Geometrical 
denote by \ the length of a luminous wave; join 0 and expknatilTj^^ 
O' by a right line, and take the distances A B = B 0 
— CD=DE—\\ and with O' as a centre and the 
distances O' B, O' 67, O' D, O' E\ Ac., successively as 
radii, describe the arcs B b, Oc, Dd, Ee , &c., cutting 
the section of the wave ME, in the points b, c, d, e, 

&c. Now, regarding the several molecules in the por¬ 
tions Ah, b c, cd, de, &c., of the great wave, as so many 
centres of disturbance, it is obvious that the secondary 
waves sent to the molecule O', from those which occupy 
corresponding positions, on each pair of consecutive por¬ 
tions, will be in complete discordance, and therefore, Joint effects of 
Acoustics, § 59, that the joint effects of any two consecu- 
tive portions will be to destroy one another, provided main wave; 


Fig. 2. 

O 



the rectilineal 
propagation of 
light in 
homogeneous 
media; 




170 


NATURAL PHILOSOPHY. 


Portions of Uie 
main wave 
remote from tlio 
straight lino 
destroy each 
other; 


Displacement of 
an assumed 
particle due to 
those portions of 
the main ware In 
the immediate 
vicinity of the 
right line joining 
it with the 
luminous origin; 


Portion 
producing the 
greatest effect; 


Effects of the 
other portions. 


the waves from these portions be equal in number and 
give equal molecular displacements. And it is easy to 
see that this is the case with respect to the portions 
remote from A. For, the magnitude of the displacement 
of O', caused by any two consecutive portions, depends— 
first, upon tbe relative magnitudes of these portions, and 
secondly, upon their difference of distance from O'. 
With respect to the former, it is ob vious, from tbe con - 
struction, that A b is greater than b c, be than c d, cd 
than d e , and so on; but that the successive differences 
go on continually diminishing, and that the magnitudes of, 
and consequently the number of waves from, the succeed¬ 
ing portions, approach indefinitely to equality as they 
recede from the point A. For corresponding points 
in consecutive portions, the difference of distance, which 
ia £ X, never exceeds, as we shall see, 0,000013 of an 
inch ; so that the portions of the main wave remote from the 
straight line O O' , destroy each other’s effects, and the 
displacement of O' , will be entirely due to those parts 
of the great wave in the neighborhood of the line con¬ 
necting the point O' with the luminous origin. 

Of these parts A b produces,' of course, the greatest 
effect, being both the largest and least oblique to O O'. 
The effects of the neighboring portions are, however, 
sensible, and we shall have occasion, under the head of 
chromatics, to observe some important phenomena to 
which they give rise. In the mean time we cannot fail 
to perceive one remarkable consequence of this explana¬ 
tion, viz.: that if the alternate portions b c, d e, &c., 
whose effects are, relatively to the others, negative, he 
stopped, the total effect upon O' will he augmented, and 
the light there will be literally increased by intercept¬ 
ing a portion of the wave. All of which we shall have 
occasion to see fully confirmed by experiment. For the 
present our conclusion is, that in a homogeneous me¬ 
dium, the apparent effects of light are propagated from 
one point to another in a right line/ that the sensible 
effects of light cannot, like those of sound, he propa- 




ELEMENTS OF OPTICS. 


171 


gated round corners, and that optic shadows must run not 
up to the right line drawn from the luminous source rounlfcliLra. 
tangent to the edges of objects which cast them.* 


§ 8. Any line R R, which 
pierces the wave surface perpen¬ 
dicularly, is called a ray of light. 
A ray, therefore, is obviously a 
line along which the successive 
effects of light occur. 

When the wave surface be¬ 
comes a plane, the rays will be 
parallel, and a collection of such 
rays is called a beam of light. 

When the wave surface is 
spherical, the rays will have a 
common point at the centre of 
curvature, and a collection of 
such rays is called a pencil of 
light. 


Fig. 3. 



Kay of light. 


Beam of light. 


Pencil of light. 


REFLEXION AND REFRACTION OF LIGHT. 

§ 9. The reciprocal action between the molecules of Beftexion an<i 
various substances and those of the ether which pervades [f^ chon <>f 
them, causes the latter fluid to exist in a state of different 
elasticity and density in different bodies. By reference to 
Ecpiation (3), Acoustics, Tve recall that the wave velocity 
increases with the elasticity of the medium and decreases 
with its density; and, § 71, same subject, shows us, that 
when a wave is incident upon the boundary of a medium 
of different density from that in which it is moving, it 
will be resolved into two component waves, one of which 
will be driven back from the bounding surface, while the Follow the sam« 
other will be transmitted and conducted through the new laws 83 sound; 
medium. Light, like sound, will, therefore, be reflected 
and refracted , and according to the same laws. 

*See Appendix No. 1. 








172 


NATURAL PHILOSOPHY. 


And the § 10. And resuming Equation (29), Acoustics, which is 

circumstances of 
incident and 

deviated light # V . f . 

determined by Sin. 9 — y/" * Sin. <p .(1) 

the same 
equation. 

we may determine all the circumstances of velocity and 
direction of incident, reflected and refracted light. In 
this equation V, denotes the velocity of light in the first, 
and F', its velocity in the second medium; 9 , the angle of 
incidence, and 9 ', that of refraction. 


Deviating 

surface; 


Incident, 
reflected, 
refracted wave.; 


Incident, 
reflected, 
refracted ray; 


Angle of 
incidence, of 
reflexion, of 
refraction. 


§ 11 . Let us Jiere repeat 
the notation of § 71, Acous¬ 
tics. The surface which 
separates the two media, 
and of which M A 7 , repre¬ 
sents a section by a normal 
plane, is called the deviating 
surface ; and, supposing the 
wave to be moving from S 
towards D , WW is called 
the incident , W W' the 
reflected , and W"W" the 
refracted wave / and the normals to these, viz.: S D, 
IJ S’ and D /S'", are called, respectively, the incident , re¬ 
flected, and refracted ray ; the ray D /S' is said to be de¬ 
viated by reflexion, andZ> S” by refraction ; also drawing 
the normal P P’ to the deviating surface, the angle 
P D /S', which the incident ray makes with this normal, is 
called the angle of incidence j the angle P D /S', which the 
reflected ray makes with the normal, is called the angle of 
reflexion , and the angle P D S’" — P’ D > S'", which the 
refracted ray makes with the normal, is called the angle 
of refraction . 


Fig. 4. 



now these angles § 12. These angles are. always estimated from that part 
are estimated, ^ norma j drawn through the point of incidence of 
the ray, which lies in the medium of the incident wave. 





ELEMENTS OF OPTICS. 


173 


They are accounted positive when on the same side of when positive 
the normal as the incident ray, and negative when on“^en 
the opposite side. Thus, 
the angle of incidence 
P DS, is always positive, 

|as also the angle of re¬ 
fraction PDS while 
the angle of reflexion 
PDS\ will always be 
negative, as it should be, 
since the velocity of the 
reflected light must be 
counted negative, the 
reflected wave being dri¬ 
ven back from the de¬ 
viating surface. 

§13. When the deviating surface is curved, we con- deviatin8 
ceive a tangent plane drawn to it at the point of incidence, 
and treat this plane as the deviating surface for that 
portion of the wave which is incident immediately about 
the tangential point. 


Fig. 5. 



Illustration. 


§ 14. The angle which 
any ray after deviation, 
makes with the prolonga¬ 
tion of the same ray be¬ 
fore incidence, is called 
the deviation. Thus, 
S lv - D is the devia¬ 
tion by reflexion; and 
S" D S lv -, the deviation 
by refraction. 


Fig. 6. 



The deviation; 


By reflexion and 
by refraction. 


§ 15. If we make 


V 

Y* = 


( 2 ) 










Equation ( 1 ) becomes 


Equation 
applicable to 
refraction; 


Equation 
applicable to 
reflexion; 


General equation 
for all deviations. 


Catoptrics and 
Dioptrics. 


Index of 
refraction; 


sin 9 = m sin 9 '.(3) 

which answers to any refracted ra}'. 

For the reflected ray, V becomes equal to — V', and 

— 1 — m\ 

this in Equation (3) gives 


sin 9 = — sin 9 '.(4) 

which applies to all cases of reflexion. And generally 
we may consider the Equation 

sin <p = m sin 9 '.(5) 

as applicable to all cases of deviation, observing to make 
m, equal to minus unity in cases of reflexion. 

§ 16. The circumstances attending the deviation of the 
component waves into which an incident w T ave is re¬ 
solved at a deviating surface, being in general different, 
gave rise to two distinct branches of optics, called 
Catoptrics and Dioptrics , the former treating of re¬ 
flected, and the latter of refracted light. But by the 
generalization expressed in Equation (5), this division 
may be avoided, the discussions made more general, and 
much space and labor saved. 

§ 17. The quantity m, is called the index of refraction. 
It is the ratio of the velocity of the incident to that 
of the deviated light, which is equal to the ratio of the 
sine of the angle of incidence to the sine of the angle 






ELEMENTS OF OPTICS. 


•175 


of refraction or of reflexion, according as m is positive 
or minus unity. 

The numerical value of m, has been determined ex- Value o{ 
perimentally for a great variety of substances, solids, under different 
liquids and gases, on the supposition that the deviating circamstances - 
surface separates the various substances from a vacuum. 

It is found to be constant for the same medium, but 
variable from one medium to another. And as a gene¬ 
ral rule, it is greater than unity when light passes from 
any medium to another of greater density, as from air 
to water, from w r ater to glass; and less than unity when 
light passes from one medium to another less dense, as 
from water to air. 

There is a remarkable exception to this rule in the case Excer , tion the 
of combustible substances, these always refracting more general rule, 
than other substances of the same density. 

From what has been said, it is obvious that a ray of Light deviated 
light on leaving any medium' and entering one more respect to 
dense, will, in general, be bent towards the normal to the the nonna ' 
deviating surface, while the reverse will be the case when 
the medium into which the ray passes is less dense than 
the other. 

§ 18. If all bodies possessed equal density, the Refractive P ow« 
value of m, or the index of refraction, might be of different 

. ° substances; 

taken as the measure ot the retractive power o£ the 
substance to which it belongs, but this not being the case, 
it has been shown, that if the expression of the law ac¬ 
cording to which all substances act upon light be of the 
same form , the refractive power will be proportional to 
the excess of the square of the index of refraction above 
unity, divided by the specific gravity. Calling 7i y the ab¬ 
solute refractive power, m y the index of refraction, A, the 
specific gravity, and A , a constant co-efficient, we shall 
have according to this rule, 

Its value ic an j 
case; 



/ 





176 


NATURAL PHILOSOPHY. 


Refractive 
indices and 
powers 

determined by 
experiment; 


The following table shows the value of m, and n, for 
the different substances named, the value of m being 
taken on the passage of light from a vacuum. 

TABLE 


Of Refractive Indices and Refractive Powers. 


Substances. 

m 

TO 3 — 1 

a 


Chromate of Lead, 
Realgar, 

_ v , , Diamond, 

Table of (41 ass flint 

refractive indices 'JldSb-nmi, 
and powers. Glass Cl’OWn, 

Oil of Cassia, 

Oil of Olives, 

Quartz, 

Muriatic Acid, 

"Water, 

Ice, 

Hydrogen, 

Oxygen, 

Atmospheric Air, 

(2,97 

(2,50 

2,55 

2.45 

1,57 

1,52 

1,63 

1,47 

1,54 

1,40 

1,33 

1,30 

1,000138 

1,000272 

1,000294 

1,0436 

1,666 

1,4566 

0,7986 

1,3308 

1,2607 

0,5415 

0,7845 

3,0953 

0,3799 

0,4528 



DEVIATION OF LIGHT AT PLANE SURFACES. 


Deviation of § 1$. Let M W, be a 

deviating surface, sep¬ 
arating any medium i?, 
from a vacuum A. A 
ray of light S D , being 
incident at _Z>, will be 
Illustration and deviated according to 

explanation; - .. ° 

the law expressed by 
Equation (3), 

sin (p = m sin 9', 


Fig. 7 . 


























ELEMENTS OF OPTICS. 


177 


w, being the index of refraction of the medium.#. The Deviation of the 
refracted ray D D\ meeting a second surface M f par- fl«trafracted 

allel to the first, and passing again into a vacuum, will be ray: 
refracted so as to satisfy the Equation, 

sin 9' = m! sin 9", 


the angle of incidence 9', on the second surface being the 
same as that of refraction at the first, and m\ the index 
of refraction from the medium B to the vacuum. But, 
in this case, Equation (2), 



Index of 
refraction from 
medium to 
vacuum; 


whence, substituting this value of m', and multiplying 
the two preceding Equations together, we obtain, 


sin 9 = sin 9", 


Operations 

performed; 


that is, the ray after passing a medium bounded by paral- 

5 J r 0 . . Couclusion in 

lei plane faces, is not ultimately deviated, but remains pa- WO rds. 
rallel to its first direction. 

The ray D" D"\ being supposed to traverse a second Same true for 
medium bounded by parallel plane faces, and of which 
the refractive index is m ", will undergo no deviation; by parallel piano 
and the same may be said of any number of media faces * 
bounded by such faces. If, now, the spaces between 
the media be diminished indefinitely so as to bring them 
into actual contact, there will still be no deviation, and 
we find that a wave will emerge from a medium, ar¬ 
ranged in parallel strata, parallel to its position before 
entrance. 


12 



178 


NATURAL PHILOSOPHY. 


To find the 
relative index of 
refraction; 


Equations 
applicable to tho 
deviations; 


Result of 
operations; 


Bale 


Example; 


Fig. 8. 


§ 20. Let us next sup¬ 
pose a ray to traverse 
two media A and By 
bounded by plane pa¬ 
rallel faces, the media 
being in contact, and 
having their refrac¬ 
tive indices denoted 
by m and m' respec¬ 
tively ; we shall have, 
by calling m", the in¬ 
dex of refraction of 

the second, or denser medium in reference to the first, 



sin y — m sin 9 ' 

sin 9 ' = m" sin 9 ".(7) 

sin 9 " = —sin 9 . 

/m T 


Multiplying these Equations together, there will result 


m — 


(7)'. 


That is to say, to find the index of refraction in the case 
of a ray passing from any one medium to another, divide 
the index of the second by that of the first referred to a 
vacuum. The index thus obtained is called the relative 
index. 

Example. What is the relative index of air and crown 
glass, the light entering the latter from the former ? The 
tabular index of crown glass is 1,52, and that of air is 
1,0003, whence 


„ 1,5200 

m ~ 1,0003 ~ 1)52 ’ 


Result 













ELEMENTS OF OPTICS. 


179 


§ 21 . If a ray pass from a 
medium to another more dense, 
the index m" will be greater 
than unity, and from equation 
(7), we shall have 

sin <p' > sin 9 "; 

and if sin 9 ' be taken a maxi¬ 
mum, or the angle of incidence 
be 90°, equation (7) will give, 

-4= sin*".(8) 

m 

from which results a maximum limit to the angle of Maximum limit 
refraction. If m" be taken equal to 1,52 for the atmos- 
phere and crown glass, 



90°; 


sin <i>" = 0,657, 


or 


<p" = 41° 5' 30", nearly; 


Example, 
atmosphere and 
crown glass; 


for air and water, m " = 1,33, and 


Atmosphere and 
water;. 


<f>" = 48° 15'; 


that is to say, the greatest angle of refraction which can 
exist when light passes from air into crown glass, is 41° 
5 ' 30"; and from air into water, 48° 15'. 

If the ray pass from a medium to another less dense, 







180 


NATURAL PHILOSOPHY. 


Light passing 
from denser to 
rarer medium; 


Angle of 
refraction taken 
# 0 °; 


Consequence; 


Analogy; 


Examples; 


Conclusion; 


Angle of total 
reflexion. 


m" will be less than unity, 
and equal to the reciprocal 
of its former value; Equa¬ 
tion ( 7 ) will then give 

sin 9 " > sin 9 ' ; 

taking the maximum value 
for sin 9 " = 1 , we shall ob¬ 
tain from the same Equation, 


sin 9 ' = 


Fig. 10. , 



(*>) 


this value for the sine of the angle of incidence, which 
corresponds to, the greatest angle of refraction when light 
passes from any medium to one less dense, is the same 
as that found before for the greatest angle of refraction, 
when the incidence was taken a maximum, in the pas¬ 
sage of light from one medium to another of greater den¬ 
sity. 

In the case of air and glass, it is 0,657; correspond¬ 
ing to an angle of 41° 5' 30"; for air and water, the 
angle is 48° 15'. 

If the angle 9 ' be taken greater than that whose sine 

is A_, the angle of refraction, or emergence from the 

m ' 

denser medium, will be imaginary, and the light will be 
wholly reflected at the deviating surface. This maximum 
value for 9 ' is called the angle of total reflexion. Light 
cannot, therefore, pass out of crown glass into air under a 
greater angle of incidence than 41 0 5' 30 " , nor out of 
water into air under a greater angle than 48° 15'. 


§ 22 . The maximum limit of refraction , and the cases 
of total reflexion , are attended with many interesting 







ELEMENTS OF OPTICS. 


181 


results. If an eye be placed in a more refracting medium Appearances due 
than the atmosphere, as that of a fish under water, it will t0 the limit of 

. , „ /» . ni. refi-action and 

perceive, by the limit ot refraction, all objects in the total reflexion; 
horizon elevated in the air, and brought within 48° 15' 
of the zenith, while some objects in the water would ap¬ 
pear to occupy the belt included between this limit and 
the horizon by total reflexion. 

Those remarkable cases of mirage , where objects are 
seen suspended in the air, and oftentimes inverted, are Those due to 
explained by ordinary refraction and total reflexion. ordmary 
The phenomena of mirage most frequently occur when total reflexion, 
there intervenes between the suspended object and spec¬ 
tator a large expanse of water or wet prairie, and towards 
the close of a hot and sultry day, when the air is calm, 
so that the different strata may arrange themselves ac¬ 
cording to their different densities. When the wind 
rises the phenomena cease. 

Fig. 1L 


Illustration, 



It is well known that in the ordinary state of the at- Apparent 
mosphere, its density decreases as we ascend; a ray of ^ 

light, therefore, entering the atmosphere at S, would un- the positions of 
dergo a series of refractions, and reach the eye at Z?, with celestial bodlc * : 
an increased inclination to the surface of the earth ; and 
would appear to come from a point, 8\ in the heavens 
above that at 8.\ occupied by a body from which it pro- 









182 


NATURAL PHILOSOPHY. 


ceeded. Hence, the effect of the atmosphere is to in¬ 
crease apparently the altitudes of all the heavenly 
bodies. % 

Relative index § 23. Dr. Wollaston suggested a method, founded on 
determined by the limit of total reflexion. to determine the relative in- 
dices and refractive powers of different substances. If 
the angle of incidence, 9 ', be measured by any device, 
Equation (9) will give, 


m 


tt _ 


1 

~—r> 
sin 9 


And thence the from which, Equation (7)', we find the absolute index, 
re rac tve power. ^ now ‘ n g t hat of air ; and the refractive power may then 
be deduced from Equation ( 6 ). 

Optical prism ; § 24. The deviating surfaces Fig. 12 . 

have, thus far, been supposed 
parallel. If they be inclined to 
each other, as M N ', we 
shall have what is called an optical 
prism , which consists of any re¬ 
fracting substance bounded by 
plane surfaces intersecting each 
other. 

Deviating planes M JV and M N\ are called the deviating planes , and 

and rciracting the angle under which they are inclined, is called the 
refracting angle of the prism. 



Deviation of a 
ray of light in 
passing through 
a prism; 


§ 25. To find the deviation of a ray of light in passing 
through a prism, Fi ff- 13 - 

let 8 D be the 
incident, D D' 
the first, and D' 

8' the second re¬ 
fracted ray. The 
total deviation 
will be 8'£8", 








ELEMENTS OF OPTICS. 


183 


which denote by 8 ; then, calling the refracting angle 
of the prism a, and adopting the notation of the figure, 
we shall have 

8 —ED D' 4 - ED' D — $ — q/ -f* — 4 / == 9 +4- — (<}/ + 4/ ) 

Equations; 


I80 0 or«=b+ MDD'+MD'I> = a+^- 


or 


a = 4/ -+• 9' 


( 10 ) 


Refracting angle, 


hence 


r'■ 


8 — (p 4- 4 1 — cl 


(IX) Deviation; 


The deviation of a ray of light in passing through a 
prism, is, therefore, equal to the sum of the angles of in- Rule. 
cidence and emergence , diminished by the refracting angle 
of the prism. 

The refracting angle for the same prism being con- Deviation for 
stant, the deviation will depend upon the angles of i n - same P rism 

7 J. J. o depends upon. 

cidence and emergence. 

Now, from Equations (11), (10), (3), and 

sin 4* = m sin 4 /,.(3)' 

by a simple process of the calculus, or by trial, it may 
be shown, that when the angles of incidence and emer- condition for 
gence are equal, the deviation will be a minimum, or™ 1 ®*™™ 
the least possible.* 

Making 9 equal to 4'j in Equations (11) and (10), we 
find, 


*See Appendix No. 2. 






184 


NATURAL PHILOSOPHY. 


Ito nee; 


Formula for 
refractive index. 


Application of 
the formula. 


Incident ray 
normal to first 
surface; 


Consequences; 


Final result. 


P = i (“ + s ) 


which substituted in Equation (3) give/ ^, 'vV v 

1 ' 




4 


_ sin * (a + i). 

171 — -.--- > 

SHI \ a 


( 12 ) 


we have, therefore, only to measure the deviation when 
a minimum, to find the index of refraction of the me¬ 
dium of which the prism is made, supposing its re¬ 
fracting angle known. 

This furnishes one of the best methods by which the 
refractive powers of different substances may be found. 
If the substance be a liquid, we may unite two plane 
glasses, making any angle with each other, by means 
of a little cement along their edges, and place the 
liquid between them where it will be held in sufficient 
quantity by capillary attraction. 

§ 26. When the ray is incident at right angles upon 
the first surface, we have, 

9 = 0, 

<P'= 0 , 

and from Equations (10) and (11), there result, 

$ = 4, — a , 

a = -vj/ 


whence 

sin (a -f 5) = m sin a . . . . ( 13 ) 

Deviation at plane surfaces by refraction, will be again 
referred to in a subsequent part of the text. 





ELEMENTS OF OPTICS. 


185 



Fig. 14. 


§ 27. Let MJST, 

MN\ be two plane 
reflectors, meeting 
in a line projected 
in M ; S D, a ray 
incident at the 
point Z>, and con¬ 
tained in a plane 
perpendicular to 
the intersection of 
the reflectors; this 
ray will be devia¬ 
ted at the point 
of the first reflec¬ 
tor, again at the point D\ of the second, and so on. 

Required the circumstances attending these deviations. 


Call the first angle of incidence (p, 


second,.<p 3 

third.<p 3 

&c., 

n th .<p w 


In the triangle P D D\ the angle at P is equal to 
the inclination of the reflectors, which denote by i , and 
we shall have 

9, “ <P 2 = h 
<P 2 - <P 3 = h 
<P 3 “ 9 4 = h 


9 » — 2 “ 9 n -, = h 

- 9 , = *; 

and by addition, 

=»- 1 • * 

<P» = <P, -n-l .i . . . . (15) 



Deviations of a 
ray of light by 
two plane 
reflectors, the 
plane of 
incidence being 
normal to their 
intersection; 


Notation; 


Equations from 
the figure; 


Sam of these 
equations; 












186 


NATURAL PHILOSOPHY. 


If 9 i bo a 
multiple of i; 


If 9 x be any multiple of i , as n — 1 . i, 


9! — w — 1 . i = 0,.(16) 


The ray will that is to say, the nth incidence will be perpendicular 

[teei™ UP ° n 1 ° ^e reflector, and the ray will, consequently, return 

upon itself. 

Example ls£. Suppose the angle made by the reflec- 
Exampiefirst; j. org an( ^ ^he £ rgt an gi e 0 f incidence, or 9 j = 60°; 

required the number of reflexions before the ray retraces 
its course. 

These values in Equation (16), give, 


Data; 60° - U ~ 1.6° = 0 


or, 

Result n = 11. 

Example 2d. The angle of the reflectors being 15°, 
Example second; and the first angle of incidence 80°, required the fourth 
angle of incidence. 

These values in Equation (15), give 


9 4 = 80° -4-1.15°. 

Result 9 4 = 35° 


Tf (f> x be not a multiple of i, there will be some value 
multiple *©™* 4 for n that will make n — 1 . i, greater than 0 ,, in which 
case, <p x — n — 1 . i, will be negative; that is, at the n th 
incidence, the ray will be on the opposite side of the 
The ray win not perpendicular. It will therefore return, but not, as before, 

return by the 
same path; 


return by the , ,, , 

by the same path. 









ELEMENTS OF OPTICS. 


187 


Example 3 d. The angle of the reflectors being 7°, the Exam P le third ; 
first angle of incidence 69°, required the number of 
reflexions before the ray returns, and the first angle 
of incidence of the returning ray. These values in 
Equation (15), reduce it to 


= 69° — n — 1 . 7° = 76° — 7° . n. 


If n = 10, 


h = 76° — 70° = 6°. 


Suppositions; 


If n = 11, 


<p n = 76° — 77° = — 1°. Result. 

or the ray begins to return at the eleventh incidence and 
the angle of incidence is 1°. 

It is obvious that the angle of incidence of the return¬ 
ing ray will increase at every deviation; there will, there¬ 
fore, be some value of the increased angle which will 
either be equal to or greater than 90°. In the first case, 1{emark8> 
the ray will be reflected by one of the reflectors into a 
direction parallel to the other, and in the second, this last 
reflexion will give the ray such a direction that it will 
meet the other reflector only on being produced back. 

§ 28. Adding the first two Equations in group (14), we 
have 

Angle made by 
the incident ray 
and the same ray 
after two 
reflexions; 

s&D' = 21 


<t>, - (p, =2 i, 
or 


That is, the angle made by the incident ray and the 




188 


NATURAL PHILOSOPHY. 


Equal to double same ray after two reflexions, is equal to double the an- 
the angle made 0 f the reflectors. It follows, therefore, that if the 

by the reflectors. . . 

angle of the reflectors be increased or diminished by giv¬ 
ing motion to one of the reflectors, the angular velocity 
of the reflected ray will be double that of the reflector. 
Application of This is the principle upon which reflecting instruments 
this principle. f or ^ measuremen t 0 f angles are. constructed. 


DEVIATION OF LIGHT AT SPHERICAL SURFACES. 

Deviation of § 29. Let MD 0 N, be a section of a spherical surface 

light at spherical 

surfaces; separating two me¬ 
dia of different den- ^ 15 - 

sities, as air and ^ 
glass, having its cen¬ 
tre at C, on the line 

Illustration and 0 C', which will be 

notation; ^ed the axis of 

the deviating sur¬ 
face ; FD a ray of 
light, incident at D, 
and D S, the direction of this ray after deviation, which 
being produced back will intersect the axis at F'. The 
point 0 , where the axis meets the surface, is called the 
vertex , which will, for the present, be taken as the origin. 

Vertex. Call F D, u; F' D, v!; CD,r; OF',/'; 0 F, f ; 
and the angle 0 C D, 0. 

Kuie first; ISTow, distances estimated in the direction of wave pro¬ 

pagation, from any origin whatever, are always negative; 
those estimated in the contrary direction, positive. 

Buie second. And, when light is incident on a concave surface, the 

radius of curvature is always positive ; when incident on 
a convex surface, negative. 

In the triangle CDF, we have the relation, 








ELEMENTS OF OPTICS. 


189 


and in the triangle CDF', 

sin 6 _ u 1 

sin 9 ' — r * 


These combined with 


sm cp = m sin 9 , 


give 


mu.{f -r) —u’ (f- r) . . . 


Equations from 
the figure; 


Combined with 
the general 
equation of 
deviation; 


• ( 17 ) 


The first of these triangles will also give, 

U 2 = ( f - r) 2 + r* + 2 ( f — r) .r . COS & Other equations 

Vt/ 7 \tf J from the figure; 

and the second, 

u' 2 = (f r — r) 2 + r 2 + 2 (/' — r).r. cos 6. 

These latter Equations by reduction become, 

=/* - 2 r(f-r ). versin S ; xh« latter 

u' 2 =f' 2 — 2v (f— r) . Yersin 6. reduced; 

Denoting the versin 0 by 2 , and eliminating u and u\ 
between these equations and Equation (17), there will 
result, 


{f— / r)-Vf ,2 —2r(f' — r).2=m(f'—r).Vf 2 —2r(f—r).Z (18) General equation 

for finding the 
intersection of 

This is a general Equation for finding the intersection deviated rays 
of deviated rays with the axis. The relation between / 1 Wlththeax * s ‘ 
and f' is somewhat complicated, and it is obvious that 
if f be made constant, the value of f' will vary for dif¬ 
ferent values of 6 ; that is to say, if a pencil of rays pro- 
ceed from a point on the axis , they will , after deviation , indefinite size. 
meet the axis in different points , depending upon the dis¬ 
tance of the point of incidence from the vertex. 






190 


NATURAL PHILOSOPHY. 


SMALL DIRECT PENCIL. 


“ ct § 30. A pencil of light having its central ray coincident 


with the axis of the 
deviating surface, is 
called a direct pen¬ 
cil ; and if such a 
pencil be taken very 
small, the quantys, 
in Equation (18), 
will be so small that 

General equation the products of 
made applicable 
to this case; 


Fig. 16. 



Equation for i 
small direct 
pencil; 


which it is a factor 
may, without mate¬ 
rial error, be omitted. This will reduce Equation (18) to 


or 


— - 

J ( on. — 


mrf 


(m-l).f+r 
and taking the reciprocal, 


Reciprocal of the 


If f be con¬ 
stant, or the 
rays all proceed 
from the same 
conclusion for a point F On the 

email direct . , - -» 

pencil. axis oefore de- 



(19) 


( 20 ) 


motion , f will 
also be constant 
for the same 
medium and 
curvature, and all the rays after deviation will meet in 











ELEMENTS OF OPTICS 


191 


some other point F' on the axis. The first of these points Radiant and 
is called a radiant, and the second a focus j and because f ° cn8 ’ 
of the mutual dependence of these points upon each 
other with respect to their positions, they are called 
conjugate foci, and the distances f and f, are called 
conjugate focal distances. The radiant is a point common f 0C ai distances ; 
to the undeviated, and a focus to the deviated rays. Then, 
a radiant is the centre of curvature of the undeviated 
wave; and a focus of the deviated wave. When a wave 
turns its convexity to the front, its molecular living force 
becomes more and more diffusive as the wave progresses ; Real and virtu&i 
when it turns its concavity to the front, more and more fodl * 
concentrative. A radiant is real, when the undeviated 
wave turns its convexity to the front; and virtual , when it 
turns its concavity to the front. X focus is real, when 
the deviated wave turns its concavity to the front; and Real and virtual 

. • i /» radiants. 

virtual , when it turns its convexity to the front. 

§ 31. Luminous waves, like waves of sound, Acoustics, Living force or 
§ 53, become more and more diffused in proportion as intensity of light 
they recede further and further from the place of primi- decreases for 
tive disturbance, provided their convexities continue to be 
turned to the front, and more and more concentrated converging raja, 
after they have been so deviated as to turn their con¬ 
cavities to the front. In other words, the living force of 
the wave molecules, which determines the intensity of 
light, will become less and less for divergent, and greater 
and greater for convergent rays. 

That portion of the living force imparted to the ethereal Living force of 
molecules at any one place, as a radiant, and which proceeds ft 

upon a spherical segment embraced by the bounding rays segment 
of a small direct pencil, can, therefore, Equations (19) and Into 

(20), be concentrated upon the ethereal molecules at 
another place, as a focus, by the action of a spherical devi¬ 
ating surface ; and the focus, whether real or virtual, be- 

’ . And the focus 

comes a source of light as well as the radiant, and is as becomes a source 
distinctly visible. When the focus is real, the deviated of H « hL 
wave first becomes concentrated in, and subsequently 





192 


NATURAL PHILOSOPHY. 


Whence the 
deviated wave 
proceeds for real 
and for virtual 
foci. 


First deviated 
ray incident upon 
a second surface; 


Equation 
applicable to the 
second deviation; 


Second deviated 
ray incident Upon 
a third surface; 


Equation 
applicable to the 
third deviation; 


emanates from it; when virtual, the deviated wave pro¬ 
ceeds only from the deviating surface, hut with dimen¬ 
sions the same as though it had departed from the virtual 
focus. 



§ 32. If the ray Fig - 18 

which is deviated at 
the first, be incident 
upon a second sur¬ 
face M' -ZP, having 
a radius r\ and 
situated at a dis¬ 
tance £, from the 
first, measured on 
the axis, we may 
suppose this ray to have proceeded originally from F '; 
and denoting the distance from the new vertex 0 \ to 
the point F ", in which this ray, after deviation at the 
second surface, meets the axis, by f \ and the index of 
refraction of the second medium by m\ we shall have 
from Equation (20), 


1 _ m' - 1 , 1 

f"~mW + m'(f + t) 


( 21 ) 


Fig. 19. 



And by the same process for a third deviating surface, 


1 _ m" - 1 1 

m"r" + 


. . (22) 












ELEMENTS OF OPTICS 


193 


Pig. 90. 



N' 


Third deviated 
ray incident upon 
a fourth surface; 


? 


And for a fourth, 


1 _ {m m - 1 ) 1 

“ m'" r'" + m'" (/"' + t") 


(23) Equation 


applicable to the 
fourth deviation, 
and so on. 


And so on for any number of surfaces, the law being ma¬ 
nifest. 

§ 33. The value of f + t, deduced from Equation (20) Direct relation 
and substituted in Equation (21), will give a direct rela- found bctween 
tion between / and /, m terms of r, r , ra, w! and t; distance and finai 
and the value of f" + t’ found from this derived equa- ? ocal distance - 
tion and substituted in Equation (22) will give a direct 
relation between f" and f\ in terms of r , r\ r ", m, m', 
m!\ t and t'; and by the same process of elimination a 
direct relation may be found between the radiant distance 
f and the final focal distance 

§ 34. But in practice the distance t , is so small that it Practical relation 
may, without sensible error, be neglected. Omitting t, ||^^ g those 
we shall find that the first member of each of the preced- omitting t\ 
ing equations becomes a factor in the last term of the 
second member of that which immediately follows it, and 
proceeding to eliminate these factors by their values, we 
obtain from Equations (20) and (21) 


1 m’ — 1 
f 77 ~ m! r' 



m — 1 ^ 1 
mr mf 


}; • • (24) 


Eesulting 
equation for 
two surfaces, 











194 


Relation 
between 
conjugate focal 
distances for 
three surfaces, 
omitting t ; 


Same for four 
surfaces, and so 
on. 


Medium between 
second and third, 
&c. surfaces, 
supposed the 
same as that of 
incident light; 


Corresponding 
values of 
refractive 
indices; 


Resulting 
equations for 
two, three, four, 
&c. surfaces. 


NATURAL PHILOSOPHY. 


this value of 



substituted in Equation (22), gives, 

i 


4,=-^+4] +-^H (25) 

j m r m { m r m \ m r mfi / ) 


and this value of 



in Equation (23), gives, 


1 

fff 

m — 


m'" r‘ 

1 r 

m"—l 

m"' L 

.m"r" * 


+ 


1 im'—l 1 /m—1 


m 


\ m ~ 1 l 1 ( 

( m! r' m'\ 


m r 



(26) 


and so on for additional surfaces. 


§ 35. If we now suppose the medium between the 
second and third , fourth and fifth, sixth and seventh , 
&c., deviating surfaces, the same as that in which the 
light moved before the first deviation, we shall have 
the case of a number of refracting media bounded by 
spherical surfaces, situated in a homogeneous medium, 
such as the atmosphere, for example, and nearly in 
contact. Hence, 


m’ = JL: = m'""-. 


m 


m 


Tt > 


m 


utr 


, &C. 


and the foregoing Equations reduce to 


~={m- l).j- -4[ + 1 

j 1 r r 1 fi 


f 


m ”—1 
m" r" 


l /I 1 

\ 1 ) 

I' 

1 

T—( 
1 

S 

H-7* f 

l \7* v 

’ /> 


(27) 


1 i 

f 1 

l \ 


J },ni ^ 1*1 

\7' 


( 7 - 7 ) 


&c., &c. 


/ 
























ELEMENTS OF OPTICS. 


195 


§36. Any medium bounded by curved surfaces and Lens defined; 
used for the purpose of deviating light by refraction, 
is called a lens. Equation (27) relates, therefore, to the 
deviation of a small pencil of light by a single spheri¬ 
cal lens; f denoting the distance of the radiant, and E( i uati0Ils 
f'\ that of the focus from the lens. Equation (28), re- 0 ne, two, &c. 
lates to the refraction or deviation by a single lens and lenses - 
a second medium of indefinite extent bounded on one 
side by a spherical surface nearly in contact with the 
lens. Equation (29), relates to deviation by two spheri¬ 
cal lenses close together, f and f nn denoting, as before, 
the radiant and focal distances. 


§ 37. If the rays be parallel before the first deviation, incident rays 

^ supposed 

f will be infinite, or —- = 0, and Equations (20), (27), paraUel; 

J 

(28), and (29), will reduce to 


1 _ m—1 # 
f'~~ mr ’ 

1 -- /I 

jrr = m -1 • 




1 m "-1 , 1 r- /I 

f" ~ m"r" + m" l m ~ l ' \ r 
JTTn = ml'— 1 Ijj" ) +m,— l 

Ac., Ac. 



Resulting fo*m 
of the preceding 
equations; 


The values of f\f , \f n \f m \ &c., deduced from these Principal focal 
Equations, are called the principal focal distances , being di8tance ' 
the focal distances for parallel rays. Denoting these 

distances by F„, F m , F w „ &c., and (J - J), (J - f) 

Ac., by —, i-, _-L-, Ac., we shall have the following table, 

P P P 

viz. : 









196 


NATURAL PHILOSOPHY. 


Table of 
reciprocals of 
principal focal 
distances; 


1 _ m—1 

F t m r 
1 m—1 

= ~r 

_1 _ m"~ 1 _ 1 _ Im- 1 

F ~ m" r" m' \ p . 


Ptt) 


m”—l m—1 


HU 

1 


m 


P 

nn _ 


F, 


urn 

1 


1 lm"—l m— 1\ 

+ ^\7 r “ +_ 7/ 


_ m —1 m’ 
F ~~ o"" + 

HUH Y 

&C., &C., &C. 


Jt ^ mr 


. (30) 


An examination of the alternate formulas of the above 
table, beginning with the second, leads to this result, 
viz., that the reciprocal of the principal focal distance of 
any combination of lenses , is equal to the sum of the re¬ 
ciprocals of the principal focal distances of the lenses 
taken separately; which may be expressed in a general 
way by the Equation, 


Value for the 

reciprocal of th® 

principal focal 

distance of any 

combination of w J}grein 

lenses. 




, denotes the reciprocal of the principal 


focal distance of any one lens in the combination, the 
Greek letter 2, that the algebraic sum of these is to be 


taken, and ——, 
F 


the reciprocal for the combination. 


First members of Substituting the first member of the first Equation, 

group (30) & 1 

substituted in in group (30), and the first members of the alternate 
equations - Equations, beginning with the second, for their corres¬ 
ponding values in Equations (20), (27), (29), &c., we 
finally obtain, 















ELEMENTS OF OPTICS. 


197 


i=JL + J_ 
/' F, + »/■ 

Jt_1 1 

f" ~ F„ + f 


JL_1_ 1 

/"" ~ F m , + / 


/ 


= -i _p — 

mm p> ' -f 

mm J 


(32) Resulting 

equations for the 


(38) 

(34) 


discussion of the 
deviation of light 
by one or more 
lenses or by a 
single surface. 


(35). 


Equations (33), (34), and (35), are of a convenient form 
for discussing the circumstances attending the deviation 
of light by refraction through a single lens, or a com¬ 
bination of lenses placed close together; and Equation 
(32), the deviation at a single surface. 


§ 38. The several terms of these Equations are the re-To find relative 
ciprocals of elements involved in the discussions which uieasures f ° r the 

vergency of 
incident and 

Fig. 21. 


are to follow. The 
pencil of light being 
small, the versed sine 
of half the arc DD\ 
has been disregard¬ 
ed, and the arc itself 
may be regarded as 
coinciding with the 
tangent line at the 
vertex 0, and as 

having been described about either of the points G\ F 
or F, as a centre, indifferently; and denoting the length 
of the arc O D by a, and the number of degrees in this 
arc when referred to the centre F, corresponding to the 
radius /j by n , we shall have the proportion, 


deviated rayri; 



Rays supposed to 
diverge both 
before and after 
deviation, and 
arc taken; 


2 x .f: 360° :: a : n / 


n 


a . 360° 1 

2* V 


Number of 
degrees in this 
arc referred to 
the centre F\ 


whence, 












198 


NATURAL PHILOSOPHY. 


in which * denotes the ratio of the circumference of a 
circle to its diameter. 

When this arc a is referred to the centre F\ corres¬ 
ponding to a radius f\ its number of degrees, denoted 
by n\ becomes, 


Number of 
degrees in same 
arc referred to 
the centre F ‘; 

and dividing the first of these Equations by the second, 
we find, 


360 c 


2 it 


1 

" f' 


Katio of the 
above values; 


1 

n __ f 
ri * 

f 


Conclusion for 
diverging rays. 


Conclusion for 
converging rays 


whence we conclude that JL and JL measure the relative 

J J 

divergence of the incident and deviated rays. 

When the devi¬ 
ated rays meet Fis * 22 ‘ 

the axis at F\ on 
the opposite side 
of the deviating 
surface from the 
radiant, the value 
being laid off 
in a contrary di¬ 
rection from the 
i vertex <9, becomes negative, and the relative measure 

for the convergence of these rays will be negative. 
Again, if the incident rays converge to a point A 7 , 
before deviation, f for the same reason, would be ne¬ 
gative, and the measure for the corresponding conver¬ 
gence would be negative. And, generally, we shall find 
that, referring the radiant and focal distances to the 






ELEMENTS OF OPTICS. 


199 


vertex as an origin, di¬ 
vergence will be mea¬ 
sured by a positive and 
convergence by a nega¬ 
tive quantity; and for 
convenience we shall, 
therefore, hereafter em¬ 
ploy the general term 
vergency to express either 
of these conditions of the 
rays, indifferently. 


Fig. 23. 



General rule for 
vergency of rays. 


§ 39. The power of a lens is its greater or less capacity Pmver of a lens 
to deviate the rays that pass through it. 

In Equations (33), (34), (35,) &c., _L -L, J_, &c., 

J1 ■*//// llllil 

will measure the vergency of parallel rays after devia¬ 
tion ; and as these measures are expressed in functions 


&c., 

they will be constant for the same media and curvature, 
and may be employed as terms of comparison for the 
other two terms which enter into the Equations to which 
they respectively belong. 


of the indices of refraction, and or (— -- \ 

p \r r' 1 


From what has been said, it is apparent that 

F 


Equation (31), will measure the vergency of parallel rays 
after deviation by any combination of spherical lenses 
whatever, and will consequently be, the measure of the 


power of the combination / and as 



is the measure 


Measure of the 
power of a lens 
or combination 
of lenses; 


of the power of any one lens of the combination, we have 
this rule for finding the power of any system of lenses, 
viz.: Find the power of each lens separately , and take the EuIe - 
algebraic sum of the whole. 


§ 40. It will be convenient to express the relation in 
Equations (32), (33), (34), &c., by referring to the centre 





200 


NATURAL PHILOSOPHY 


To find a relation 
between the 
conjugate focal 
distances when 
the centre of 
curvature is 
taken as the 
origin; 


Substitutions and 
redactions; 


Relation for one 
surface; 


For a second 
surface; 


For a third 
surface; 


Relations for a 
lens, Ac. 


of curvature of the deviating surfaces as an origin. .For 
this purpose, let 0 D 

, ,. - Fig. 24 

be a section ot the 
deviating surface, 
and denote the dis¬ 
tances of the radiant 
and focal points from 
the centre C, by c 
and d, respectively; 
we have by inspection, 

f = r + c, 
f'—r + c', 

which in Equation (19), give, after reduction, 



1 _ m — 1 
d ~ r 


m 

c 


(36) 


and for a second deviating surface whose centre of curva¬ 
ture is at a distance tf, from that of the first, we ob 
tain from Equation (36), 


1 inf - 1 m! 
c" “ r' + d + t 


. • (37) 


and for a third, whose centre is at a distance t\ from 
that of the second, 



m' 


d'+t' 


. . (38) 


c' being eliminated between Equations (36) and (37), a 
relation between c and c", will result; in like manner, 
c" being made to disappear by means of this derived 
equation and Equation (38), there will result an equa¬ 
tion in terms of d” and c, and so for others. 


41. Retaining the thickness t y of the medium between 










ELEMENTS OF OPTICS. 


201 


the two deviating surfaces to which Equations (19) and Retaining for 
(21) relate, we obtain from the first by adding t , to both tvvosurface8; 
members and reducing to a common denominator, 


ft + t __ mrf + (m— 1 ./ + r) t 
m — 1 . f d- r 

and this substituted in Eq. (21), at the same time making And 8 U p pose 
m =-, which is supposing the ray to pass into the Wd a mediura 

^ immersed in 

first medium after having traversed the medium bounded another; 
by the two deviating surfaces, that Equation reduces to, 


1 1 — m m(m — 1 .f + r) Final relation for 

wT = —? -1---r . -— * * (p y ) a single lens 

T r mrf + ( f.m -— 1 + r) t retaining t. 


which gives a direct relation between the conjugate focal 
distances in the case of light deviated by a single lens. 


APPLICATION OF THE PRECEDING THEORY TO THE DEVI¬ 
ATION OF LIGHT BY REFRACTION THROUGH THE VARI¬ 
OUS KINDS OF SPHERICAL LENSES. 


§ 42. A lens has been defined to be, any medium Application of 
bounded by curved surfaces, used for the purpose of 
deviating light by refraction; the surfaces are generally various spherical 

■, . , lenses. 

spherical. 


A, called a double 
convex lens, is bounded 
by two spherical sur¬ 
faces, having their cen¬ 
tres and the surfaces 
to which they corres¬ 
pond, on opposite sides 
of the lens. When the 



Fig. 25. 




Geometrical 
representations 
of the spherical 
lenses. 















m 


NATURAL PHILOSOPHY. 


Double convex 
lens; 


Plano-convex; 




curvature of the two 
surfaces is the same, 
the lens is said to be 
equally convex. 

E, is a lens with one 
of its faces plane, the 
other spherical, this 
latter face and its cen¬ 
tre being on opposite 

sides of the lens, and is called a plano-convex lens. 

Double concave; c, a double concave lens / each curved face and its 
centre lying on the same side of the lens, 
piano-concave ; _Z), is a plano-concave lens , having one face plane and 

the other concave. 

E\ has one face concave and the other convex, the con- 
Memscus; vex face having the greater curvature; this lens is called 
a meniscus. 

E, like the meniscus, has one face concave and the 
other convex, but the concave face has the greater 
curvature; this is called a concavo-convex lens. 

The line containing the centres of the spherical 
surfaces, is called the axis. 


Concavo-convex. 


Different cases § 43. A moment’s consideration will show that all the 

arise from the circumstances of vergency attending the deviation of light 

signs of the radii; 0 " # 0 o 

by any one of these lenses, will be made known by 
Equation (33), it being only necessary to note the dif¬ 
ferent cases arising out of the various combinations of 
surfaces by which the lenses are formed; these cases de¬ 
pend upon the signs of the radii. 

Equations (33), (34), (35), &c., were deduced on the 
Eu.©for signs of supposition that r is positive, the concave side of the 
surface being turned towards incident light; it will, of 
course, § 29, be negative when the convex side is turn¬ 
ed in the same direction. Besides, f was taken positive 
for a real radiant , or when the rays are supposed to di¬ 
verge from any point upon the axis of the lens, before 
deviation; on the contrary, it will become negative when 





















ELEMENTS OF OPTICS. 


203 


the rays are received by the deviating surface in a state And of conjugate 
of convergence to a point behind the ‘lens. The signs f ° cal dl!,tancc8 ' 
of /', /", &c., will be positive when the deviated rays 
meet the axis on being produced back. The foci are then 
virtual. When the rays meet the axis on the opposite 
side of the lens or lenses, f'\ &c., become negative, 
and will correspond to real foci. 

The several lenses may be characterized as follows : 


1 Double Convex , 


21 


3 


— r and + r* 

— r and + r'= oo 


Plano-Convex , convex side to in¬ 
cident light, 

Do. plane side to inci¬ 

dent light, . . -j- x == go and 4- r' 
Meniscus , convex side turned to 

incident light, . . T <^r\ — T, — T* 

_ Sa?ne, concave side do. do. -f- 7*, -f* r' 

4 Double Concave ,. 4“ r , — V* 

Plano-Concave , concave side to 

incident light, . -f- 7*, 4“ v’ — 0° 

u Same, plane side to do. do. -fr,= 00 , and— T* 

Concavo-Convex , concavo side to 

incident light, 4* V* 

Same , reversed, '. . r > r\ —r 1 ,— v' 




6-1 


Characteristic* 

(^1) ^ ie var * ou * 
lenses. 


ft 44 . To discuss the properties of any one of these Discussion of th* 

^ properties of any 

lenses, resume i en s. 

Equation (33), de- Flg ' ~ 6 ’ 

termine the sign 

of -jr> b y refe ' 

rence to its gen¬ 
eral value in 
Equations (30), 

and the table above, and then proceed to make various 
suppositions in regard to the position of the radiant and 
deduce the corresponding places of the focus. 
















204 


NATURAL PHILOSOPHY. 


Double convex 
lens taken as an 
example; 


General 
equation; 


Value for 
reciprocal of 
principal focal 
distance; 


Equation for 
discussion; 


Real radiants 
between 
principal focus 
and infinity; 


Give real foci. 


Real radiants 
within the 
principal focus; 


§ 45. As an example, let us take the double convex 
lens. 

Equation (33), is 


1 

/" 


± + 1 
F, f 


and, Equation (30), and Table (A), 
1 m — 1 


F„ 


P 


— — m —1 (— 
\ r 


and as long as m > 1, we shall have, 




J_ + L 

F„ + / 


(40) 


For __ > or / > F u , f" will be negative, and 

J 

the vergency of 
the deviated 
rays will be ne¬ 
gative. That 
is to say, if a 
wave proceed 
from a point 
upon the axis in front of the lens between the limits F ;/ , 
the principal focus, and infinity, it will be concentrated 
after deviation, into a point upon the same line behind, 
and the focus will be real. 
t? 1 1 

* 0r p~ < yr 5 01 ' Fig. 28. 

/ < F„, f” will be 
positive, and the ver- 
gency of the deviated 
rays will be positive. 

That is, if the wave pro¬ 
ceed from a point in 
front and situated be- 



Fig. 27. 









ELEMENTS OF OPTICS. 


205 


tween the lens and the principal focus, it will, after Giv0 vi r tual f( > cl 
deviation, proceed from some other point in front, and the 
focus will he virtual. 


For 


/ 


U. = 0, or /" = oo 


Fig. 29. 



Eeal radiant at 
principal focus. 


4r = 4> or / = ^> 

•*// f 

|That is, the vergency 
of the deviated rays 
will be zero, and a 
spherical wave pro¬ 
ceeding from the prin¬ 
cipal focus will be con¬ 
verted, by deviation, 
into a plane wave which 

can only be concentrated into a point at an infinite 
distance. 

If the rays be received by the lens in a state of con- For virtual 
vergence to a point behind, that is, if the concavity 
of the wave be turned to the front before deviation, 

then i or f will be negative, and Equation (40), becomes 

it 


f \ 

\ 

1 

1 

\ 

ryn 

1! 

1 __ 1 _ 

ill 

.up 

1 

ill 

{ m 

m 


radiants; 


f" ~ — /)’ 


The equation 
becomes; 


Fig. 30. 




-And the foci will 

always be real. 


and the vergency 
of the deviated rays 
will always be ne¬ 
gative. In other 
words, to whatever 
point behind the 
lens the wave may 
be tending to con¬ 
centration before de¬ 
viation, the deviation will cause it to concentrate in some 
other point behind. 

If the rays proceed from a point in front and at the ^ _ ir . lt . 

distance of twice the principal focal distance, / becomes 2 f„ ; 
equal to 2 F u , and Equation (40) reduces to 



Eeal radiant at 







I 


206 


NATURAL PHILOSOPHY. 


The focus will be 
real and at a 
distance behind 
the lens eqnal to 
2 F„. 


JLJL_ _1_ 

f" F„ + 2 ~ 2 F* 


Fig. 27. 



same distance behind the lens. 

Divergence of ]? or a p cases of positive vergency, both before and 

rays decreased; A 

after deviation, we find 


r < /’ 

which shows ns that a positive vergency will be dimin¬ 
ished by the deviation. 

And convergence ]? or a p cases of negative vergency, we find numerically 

increased | 

/" ^ /’ 

but algebraically, 


1 _ ± 
r < /' 

Hence the effect That is to say, when the rays diverge before deviation, 
of a convex lens they will diverge less after; and when they converge be- 
—^ fore deviation, they will converge more after. Hence we 
conclude, that the tendency of a convex lens is to collect 
the rays, or concentrate the waves of light deviated by it. 

The focal distance of the double convex lens is given 
by Equation (27), 






ELEMENTS OF OPTICS. 


207 


/" 


f.r.r' 


Focal distance; 


rr' —f (m — 1) (r + r') 

If the lens be supposed of glass, m = §, nearly, and 


/"= 


2 frr f 


— 2 r r' 

If the lens be equally convex, r — r\ and 

f” _/• r . 

f—r 


For lens of glass; 


For lens equally 
convex; 


and if the rays be supposed parallel before deviation, 
/= oo, and 


/"= 


— r. 


For parallel rays, 


§ 46. Each of the other lenses described may be sub¬ 
jected to a similar discussion. This being done, the re¬ 
sults will conform to those exhibited in the following 


TABLE 


Lens. 


Incident pencil. 


' | Diverging | 



Convex 

— F„ S 


Sign of/" 


Kefrac.pencil. 


j | / >F„ j- -j — j- -[ Converges. 
1 li . .+ . J. j Diverges 


[{ r< F " \\f>f S 1 less: 


Table for convex 
and concave 
lenses. 



\r<f 

L't/ 


H 


Converges 

more. 


t Diverges 
\ more. 


Concave 
• +K i 


( Converging 
^ ( —/ 



/ > K [ J + 


| j Diverges. 

) J Converges 

> J leas. 













208 


NATURAL PHILOSOPHY. 


Covex lenses 
collect, and 
concave lenses 
disperse the 
light. 


To construct the 
focus; 


Illustration; 


Interpretation. 


Coi\jugate foci 
supposed in 
motion; 


A similar table may also be constructed by formula 
(34), for a combination of any of the spherical lenses 
taken two and two, and by formula (35), for any com¬ 
bination taken three and three, and so on. 

In general, it may be inferred from the preceding 
table, that convey lenses tend to collect the incident rays, 
while concave lenses, on the contrary, tend to scatter 
them. 


§ 4T. Transposing, in Equation (33), -j to the first 
member, we get 

J_ _ _1_ _ J_ 

f" f ~ K ' 


which shows that the vergency after, diminished by 
that before deviation, gives a constant vergency measured 
by the power of the lens. Hence, to construct the focus , 
draw the extreme ray 
FD , and from the point 
Z>, the line D H , mak¬ 
ing with the incident 
rayi^Z*, produced, the 
angle IID K, equal to 
the power of the lens; 

D II will be the de¬ 
viated ray, and the 
point F ", where it meets the axis, will be the focus. 
For, in the triangle F D F’\ the angle D F" <9, 



measured by i, diminished by D F F ", measured 

by ~ , is equal to HD K, measured by -JL; which is 
the geometric interpretation of the above equation. 


§ 48. Suppose the conjugate foci to be in motion, 
and denote any two consecutive values of / by x and x\ 





209 


- tC r " 





// 




and the corresponding values of f" b y y and y\ then Notation and 
Equation (33), e,u “ ti0M; 



subtracting the second from the first we find, 


1 _ 

y 



Transformations 
and reductions; 


reducing to a common denominator, and writing for the 
products y y' and x x\ the quantities f" 2 and / 2 , to 
which they will be sensibly equal, the Equation becomes 

y’ — y _ x' — x % 

f"2 y 2 9 

and dividing by the interval of time t , during which Time#, 
the change from x to x' takes place, which is the same introducod; 
as that from y to y\ we have 


v' — y 1 x r — x 1 

t t T 2 


or, 

V" V 

jn, = jri .(«) 

in which V denotes the velocity of the radiant, and V" 
that of its conjugate focus; and since the denomina¬ 
tors must always be positive, being squares, the signs 
of the two velocities must always be alike. Whence 
we conclude, that in lenses a change in the place of 
the radiant will always be accompanied by a change 
of its conjugate in the same direction, and that the 
rate of change in the one will be to that of the other 
as the squares of their respective distances from the 
lens directly. This has an important application in the 
action of lenses when employed to form images. 

14 


Relation between 
conjugate focal 
distances and 
velocities of 
conjugate foci; 


Conclude, that in 
lenses conjugate 
foci always move 
in same 
direction. 








/ yyt t ^ u — l 

c 7 “ c' * r 

J. - 22± . , W.-/ 

<■" cVf r' 


(3^ 
( 3 J] 


r 


Hi r 




210 


NATURAL PHILOSOPHY. 


If the lews be a 
sphere. 


§ 49. If the lens be a sphere, m! = —> t — 0, and —r> 

o r 7 m 7 c 

from Equation (36), being substituted in Equation (37), 
we obtain 


jL _ __ 2 (m — 1 ) 1 1.(42) 

<?" m 7 * c 


§ 50. If in Equation (20), we make r infinite, we get 


Deviation at a 
plane surface by 
refraction; 

or, 


1 _ 1 
/' “ mf 

«*/ = /', 


which answers to the case of a small pencil deviated 
at a plane surface separating two media of different 
densities, as 'air and water. On the supposition that the 

Radiant in , 2 

denser medium- radiant is in the denser medium, m becomes —, and 

rn 


this in the preceding Equation gives 


/= mf; 


Illustration; 


Appearances 
accounted for. 


that is, to an eye situated without this medium, the dis¬ 
tance of the radiant 
from the deviating sur¬ 
face will appear dimin¬ 
ished in the ratio of 
unity to the relative in¬ 
dex of refraction of the 
ray in passing from the 
denser to the rarer me¬ 
dium. This accounts 
for the apparent eleva¬ 
tion above their true positions of all bodies beneath the 
surface of fluids, as the bottom of a vessel partly filled 
with water, and the apparent bending of a straight stick 
at the surface when partly immersed in the same fluid. 








ELEMENTS OF OPTICS. 


211 


APPLICATION TO THE DEVIATION OF LIGHT 
BY SPHERICAL REFLECTORS. 

§ 51. In reflexion, we have only to consider one de- E( * uation 
viating surface. Equation (20) applies here by making 6p hericai concave 

m = “ 1, Which reduces it tO, reflector; 


12 1 

/' ~ r f . 


But two cases can arise, and these are distinguished by 
the sign of the radius. The reflector may be concave 
towards incident light, in which case r will be positive, 
or it may be convex towards the same direction, when 
r will be negative. Equation (43) relates to the first 
case, which will now be discussed. 


If the incident rays be parallel, 


— — 0, and 
/ 


Incident rays 
parallel; 



Hence the principal focal distance is equal to half radius , 
and Equation (43), reduces to 



(44) 


Equation fb? 


Now, this Equation is only concerned with the re¬ 
flected wave, and if this wave be concentrated at all 
after deviation, it must be upon that part of the axis 
on the side of the incident light, and hence /', for a 





212 

NATURAL PHILOSOPHY. 

Real radiants 
beyond the 
principal focus 

real focus must be positive, and for a virtual focus ne¬ 
gative. 

As long as —L > ~ , or f > F# f will be positive, 

and the vergency of the deviated rays will be positive; 
that is, a wave proceeding from a point in front of the 
reflector between the principal focus and infinity will, 
after deviation, be concentrated into some other point in 
front. 

Real radiants 

within the 
principal focus; 

the vergency will be negative; in other words, a wave 
proceeding from a point on the axis between the vertex 
and principal focus, will never be concentrated after de¬ 
viation, but will appear to proceed from a virtual focus 
behind. 

If the radiant be at the centre of curvature, / = 2 F n 
and 

Radiant at the 

centre of 
curvature. 

/' = 2^=r; 

that is, a wave proceeding from the centre of curvature 
will, after deviation, return to that point. 

For 

Real radiants 
beyond the 
centre \ 

/> 2 F t or/ > r; 

we have 

r > A? 0If,<n ■ 

Give real foci 
between the 

centre and 
principal focus; 

or the focus will be between the reflector and centre, and 
since -L — 1 < A, we find I < A, or /' > F, \ so 

that the focus will be found between the centre and prin¬ 
cipal focus. 



ELEMENTS OF OPTICS. 


213 


If 


/ < 2-P) or/^ r; 


Real radiants 
between the 
centre and 
principal focus; 


then will 


/' <2 F,' 


or /' > r; 


Give real foci 
beyond the 
centre; 


that is, the focus will be at a greater distance from the 
reflector than the centre. 

When / = F t we shall have -L = 0 ; that is, the ver- Real radiant at 

f the principal 

gency will be zero, which shows that a spherical wave focus, 
proceeding from the principal focus will be transformed 
by deviation into a plane wave, which can only be con¬ 
centrated at a distance /' = oo . 

If the vergency before incidence be negative, / will 
be negative, and Equation (44), becomes 



(45) 


Virtual radiants • 


Hence, /' will always be positive, and the vergency Alway8givereal 
positive; that is, when a wave is proceeding to con-foci 
centration in a point behind a concave reflector, it will, 
after deviation, be concentrated into some other jioint 
in front. 


Equations (44) and (45), show that , which mea¬ 
sures the vergency of deviated rays, is always algebrai- reactors 

q # analogous to 

cally greater than _, which measures the vergency of convex lenses. 

the incident rays. Hence, concave reflectors, like con¬ 
vex lenses, tend to collect the rays of light which are 
deviated by them. 




214 


NATURAL PHILOSOPHY. 


Different cases § 52. By discussing the several cases that will arise in 
m re ec or 3 , attributing different signs to r and f, and various values 
to the latter, we shall find the results in the following 


TABLE. 


Table for convex 

Eeflector 

Incident pencil. 

„-A 


Sign of ~ji 

Deflect, pencil 

and concave 
reflectors; 

Concave 

+ F, ' 

f A 

- | Diverging £ 

U- 7 S. 

\\ />K 

M + 

\\f 7/ 

j- | Converges 

> j Diverges 

1 1 less. 



| Convening 

Iby i 


{At 

| ( Converges 
f ( more. 



- | Diverging 

l-Hf 


{/‘~t 

) 1 Diverges 
j } more. 


Convex 
— F, 

l 

( Converging 

U -/ 

H + 7^ 1 

f>F, 

f <F 

H - 
WAt 

j | Diverges. 

) t Converges 

5 i less. 


Conclusions. from which we perceive that convex reflectors tend to 
scatter the rays and concave reflectors to collect them. 


§ 53. If y, in Equation (44), be transferred to the first 
member, we find 


l 

f + f ~ F, 


Bran of the 

vergencie. after which shows that the vergency after, increased by that 
“viauoT before deviation, is a constant vergency, which is mea- 
coDstMt; sured by the power of the reflector; and to construct 









ELEMENTS OF OPTICS. 


215 


the focus, draw the F1& 34 Construction of 

extreme ray FD , and 
the line DF\ mak¬ 
ing with the normal 
DC\ the angle CDF’ 
equal to the angle 
of incidence, the 
point F\ where this 

line meets the axis, will be the focus. The reason is 
obvious. 



§ 54. By a process entirely similar to that of § 48, we For reflectors 
may find from Equation (44), which appertains equally to con j u s ate f,>cl 

^ 1 ' ' 7 rsr x J more in opposite 

a concave or convex reflector by assigning to i its pro- directlon3 - 
per sign, 


F^ __F 

f' 2 ~ f 2 


(46) 


and because F' and F have contrary signs, we conclude 
that the conjugate foci in the case of spherical reflectors 
proceed, when in motion, in opposite directions. 


§ 55. Equation (43), by making r infinite, reduces to. 


1 ^ 

r 



Deviation by 
reflexion at 
plane surfaces; 


or, 

/' = -/, 

"Which shows, that in all cases of deviation of a pencil 
by a plane reflector, the divergence or convergence will 
not be altered ; and if the rays diverge before deviation, 
they will appear after deviation to proceed from a point conclusion, 
as far behind the reflector as the real radiant is in 
front; but if they converge before deviation, they will 
be brought to a focus as far in front as the virtual radi¬ 
ant is behind the reflector. 






216 


NATURAL PHILOSOPHY. 


8phericai 
aberration; 


Incident pencil 
not email; 


Illustration; 


Longitudinal 
aberration; 


lateral 
aberration; 


SPHERICAL ABERRATION, CAUSTICS, AND ASTIGMATISM. 

§ 56. Thus far the discussion has been conducted upon 
the supposition that the pencil is very small, and that z, the 
versed-sine of the angle d, included between the axis and 
the radius drawn to the point of incidence of the extreme 
rays of the pencil, is so small, that all the products of 
which it is a factor may be neglected. If, however, z be 
retained, and Equation (18) be solved with reference to 
f, the value of this latter quantity will be expressed 
in terms of m, f\ r and z, and may be written 

and if the semi-arc of the deviating surface, denoted by 
d, and of which 
z is the versed- 
sine, be made to 
vary from zero 
to any magni¬ 
tude sufficient to 
embrace the ex¬ 
terior rays of any 
definite pencil, 
it is obvious that f’, must have an infinite number of 
values, and that each value will give the focus for those 
rays only which make up the surface of a cone and are 
incident at equal distances from the vertex. This wan¬ 
dering of the deviated rays from a single focus is called 
aberration , and when caused by a spherical deviating 
surface, as it is in the case under consideration and in 
practice generally, it is called spherical aberration. When 
estimated in the direction of the axis, it is called longitu¬ 
dinal, and at right angles to the axis, lateral aberration. 

If we represent the second member of Equation (19) 
by M, that Equation may be written 



f' = M 


(19)' 







ELEMENTS OF OPTICS. 


217 


and subtracting tbis from Equation (47), we find Measure of 

longitudinal 
and lateral 

_y' = _ ]\£ .fqg) aberration, and 

their laws of 
yariation; 

in which the first member denotes the length of the por¬ 
tion F' F z , of the axis along which the different foci will 
be distributed, and will measure the longitudinal aber¬ 
ration. The lateral aberration is measured by the length 
of the line F' Z, drawn through the focus of the rays 
near the axis of the pencil and perpendicular to the axis 
of the deviating surface. The linear length of the arc, 

O D — r . d, is called the radius of aperture, and it is found Radius of 
that in all cases of ordinary practice, the longitudinal aperture ' 
aberration varies as the square, and the’ lateral aber¬ 
ration as the cube of the radius of aperture. 

If in Equation (48), we make m = — 1, we shall have for a 

the longitudinal aberration for a spherical reflector. 

If the value of // in Equation (47), be substituted for 
/ in Equation (18), and we write f" for /', then solve 
the equation with reference to /", still retaining z and 

i i i.i _ , f Aberration for a 

take the difference between this value of / and thati en s. 
given by Equation (27), we shall find the longitudinal 
aberration for a single lens; and that for any number 
of lenses placed close together might be found by the 
same process. 

§ 57. We perceive that a spherical wave of any con¬ 
siderable extent deviated at a spherical surface, will not, 
in general, be concentrated at, nor will it appear to pro- spherica i 
ceed from, the same point; but if we conceive the wave aberration; 
to be divided into an indefinite number of elementary 
zones by planes perpendicular to the axis of the devi¬ 
ating surface, each zone will have its particular point of 
concentration or of diffusion, according as the foci are 
real or virtual. Moreover, longitudinal aberration di¬ 
minishes the focal distance, that is, in general, // is less 
than /', and the deviated rays which are in the same 
plane and on the same side of the axis, will intersect aberration ; 




2 IS 


NATURAL PHILOSOPHY. 


each other before they do this latter line. Thus, if FD 


Fig. 


Geometrical 
r^nstration; 



Explanation of 
tiie figure; 


Caustic curve; 
Caustic surface; 


Section of the 
deviated wave 
by a plane 
* through the axis 
of the surface; 


be the exterior, and FD' its consecutive incident ray, 
D F z and D' F", the corresponding deviated rays, these 
latter will intersect each other at some point as c', on the 
same side of'the axis OF; in like manner, if D" F f 
be the next consecutive deviated ray to JD' F", it will 
intersect this latter in same point as c", and so for 
other deviated rays up to that one which coincides with the 
axis. The locus of these intersections c', c ", &c., is called 
a caustic curve; and if the curve be revolved about the 
axis 0 F, it will generate a caustic surface . This surface 
will spring from the focus of the axial rays at F', as a 
vertex, and open out into a trumpet-shaped tube towards 
the deviating surface. 

The deviated wave will no longer be spherical, but will 
be of such shape that its section d'd" d’” o', by a plane 
through the axis of the deviating surface, will be the in¬ 
volute of the section c' c” F', by the same plane, of the 
caustic surface, taken as an evolute. 

If after deviation the wave approach the caustic, the 

latter w T ill be real , 
Flg-8L being formed by 

the doubling over, 
as it were, of the 
deviated wave up¬ 
on itself, thus pro¬ 
ducing at the cusp 
c' double the ethe¬ 
real agitation due 







ELEMENTS OF OPTICS. 


219 


to either segment F z o' or c r c / separately. If, on the con- virtual caustic, 
trary, the wave recede from the caustic on being devia¬ 
ted, the caustic will be virtual. Caustics are finely illus¬ 
trated on the surface of milk when the light is reflected 
upon it from the interior edge of the vessel in which Illustration - 
it is contained. 

§ 58. We Have only spoken of a pencil of light whose 0bIiquo penca; 
radiant is on the axis, which is usually called a direct 
pencil. When the radiant is off the axis, the axial ray 
of the pencil becomes oblique to the deviating surface, 
and the pencil is said to be oblique. In the case of an 
oblique pencil, however small, the deviated rays will not, 
in general, meet the axis as in the case of the direct 
pencil, but will all intersect two lines at right angles 
to each other and not situated in the same plane. These 
lines are called focal lines , and the property of the de- Focal itnw ; 
viated rays by which all of them intersect both of these 
lines, is called astigmatism. Astigmatism 

§ 59. It is, generally, not possible to deviate a spherical Aberration 
wave of sensible magnitude by a single lens or surface destroye(1, 
of spherical form without aberration, and yet the practi¬ 
cal difficulties in grinding lenses and reflectors to any 
other figure render it necessary to adhere to this shape. 
Fortunately, however, two or more lenses may be so 
united that the aberration of one shall counteract that 
of another, and light may thus be deviated without 
aberration. When such combinations are used, a wave 
proceeding from one point may be made by deviation 
to proceed from, or concentrate in, some other point. 

Such points are called ajplanatic foci, and the combi- Apianatic foci, 
nations which produce them are said to be ajolanatic. combinations. 



220 


NATURAL PHILOSOPHY. 


Oblique pencil 
through the 
optical centre; 


Explanation. 


To find the 
optical centre of 
a lens or a 
surface ; 


Relation from 
figure; 


OBLIQUE PENCIL THROUGH THE OPTICAL CENTRE. » 

§ 60. We have seen, article (19), that a ray undergoes 
no ultimate deviation when it passes through a medium 
bounded by two parallel planes. If, then, in the case 
of an oblique pencil the rays diverge sufficiently to 
cover the entire face of a lens, there may always be 
found one at least which will enter and leave the lens 
at points, where tangent planes to its surfaces are pa¬ 
rallel. This ray being taken as the axis of a very small 
pencil proceeding from the assumed radiant, will con¬ 
tain the focus of the others, the distance of which from 
the lens, in very moderate obliquities, will be measured 
by given in Equation (27). This is obvious from 
the fact that in the immediate vicinity of the tangen¬ 
tial points the surfaces, which are spherical, will be 
symmetrical in resjDect to the line which joins them. 

To find where the ray referred to intersects the axis of the 
lens after deviation at the first 
surface, let M JST N' M’ repre¬ 
sent a section of a concavo- 
convex lens, of which the ra¬ 
dius CO of the first surface is 
r, and O' O' of the second is r '; 

/S P and S' P’ the traces of 
two parallel tangent planes. 

Denote by t the distance 0 0\ 
between the surfaces measur¬ 
ed on the axis, and by 6 the distance 0 from the first 
surface to the intersection of the line joining the tangen¬ 
tial points P , P\ with the axis. Then, since the radii 
C P and C' P\ drawn to the tangential points, must be 
parallel, the similar triangles CP Wand C'P'W, will 
give the relation, 

CO _ C' O' 

CK~ C' K 


Fig. 88. 








ELEMENTS OF OPTICS. 


221 


and replacing these quantities by their values, 

r __ r ' 

r — e ~~ r '— t — e 


Same in other 
terms; 


jfrom which we find 

rt _ r 
/—r ~ r' r 


(49) 

Result. 


But this value of e is constant, whence we infer that 
all rays which emerge from a lens parallel to their di- optical centre 
rections before entering it, proceed after deviation at the defined ' 
first surface in directions having a common point on the 
axis. This point is called the optical centre , and may 
lie between the surfaces or not, depending upon the 
figure of the lens. 

If we suppose but one deviating surface, then the 
medium behind must be of indefinite extent, in which 
case r' and t will become infinite and sensibly equal, 
and Equation (49) reduces to 


e= r. 


That is to say, the optical centre of a single deviating optical centre of 
surface is at the centre of curvature. a single surface ; 

If the lens be double concave, the radius r f becomes 
negative, and the value of e, in Equation (49), becomes 

rt 

r'-hr’ 


and if the faces be equally concave, r will equal /, and 

t 

6 ~ 2 * 

Of a double 

That is, the optical centre is midway between the faces, concave lens; 








222 


NATURAL PHILOSOPHY. 


Of a double 
convex lens; 


Of a meniscus; 


Of a 

plano-convex 

lens. 


Optical images; 


Explanatory 

remarks; 


If the lens be double and equally convex, r becomes 
negative, and the result will be the same as above. 

In the case of a meniscus with its concave face turned 
towards incident light, the radii will both be positive, 
and r > r\ whence 


r t 

e — —->• 

r—r 

In a plano-convex lens having its plane face turned 
towards incident light, r will be infinite, and r' finite 
and positive, and 


e — — t. 

which brings the optical centre to the vertex of the 
curved face. The student may determine in the same way. 
the optical centre of the other lenses. 


OPTICAL IMAGES. 

§ 61. The surface of every luminous body is made up 
of a vast number of radiants, from each of which waves 
of light proceed in all directions. These waves cross each 
other; and if any deviating surface be presented, it be¬ 
comes the common base of a multitude of pencils, whose 
vertices are the radiants which make up the surface of the 
body. Some one ray of each of these pencils will pass 
through the optical centre of the surface, and those rays in 
the immediate vicinity of this one constituting a small pen¬ 
cil will be brought to a focus upon it as an axis, and hence 
for each radiant m the surface of the body there will be 
a corresponding conjugate. These conjugate foci make 
up a second luminous surface, from which waves will pro¬ 
ceed as from the original body; and this surface is called 




ELEMENTS OF OPTICS. 


223 


an image of the body , because to an eye so situated as to Image of 8 b0(]y 
receive these new waves, the object, though often modi¬ 
fied in shape and size, will seem to occupy the position of 
the new surface. 

An optical image is, therefore, an assemblage of foci optical image 
conjugate to a series of contiguous radiants on the defined; 
surface of some object; and its formation consists, in so 
deviating portions of the waves of light which proceed its formation 
from the object, as either to concentrate them in some consists in; 
new positions from which they may proceed as from . 
the object itself, or to cause them to move from these 
new positions without having at any time occupied them. 

In the first case the image will be real and in the second Eeai image; 
virtual. In general, but a part of each wave can be de¬ 
viated by the use of spherical deviating surfaces to sat¬ 
isfy these conditions, for those portions remote from the V irtuai imago, 
undeviated ray of each pencil cannot, in consequence of 
aberration and astigmatism, be brought to accurate ver- 
gency. 

§62. To ascertain the relation between an object and to find the 
its image, let us suppose the deviation to be produced 
by a lens, so thin that its thickness may be neglected, image formed 
which is the usual case in practice. The optical centre alens ’ 
may be taken 

, i . . » Fig. 39. 

as the origin ot co- 


sumed point P in 
the object, and 

writing this quantity for /, in Equation (33), tvhich we 
may do without sensible error, we get 


ordinates. Denot¬ 
ing by Z, the dis¬ 
tance from this 
point to any as- 




Cor ^ usatc 
corresponding ta 
an assumed 
radiant point. 










224 : 


NATURAL PHILOSOPHY. 


section of the Let the object be a plane, perpendicular to the axis 
be^righukie^l ens j its section will be a right line P Q. Call d, 
the angle included between the axis of any oblique pen¬ 
cil and the axis of the lens. When the pencil becomes 
direct, 6 will be zero, and l will equal /. But, generally, 
we have 


General relation; 



this in Equation (50), reduces it to 

/"=- 5 '..( 51 ) 

1 + -geos a 

which is the polar equation of the image referred to the 
optical centre as a pole. It is the same in form as the 
polar equation of a conic section, which is 


Equation of the 
image of a right 
line; 


Is the same in 
form as that of a 
conic section; 

Conclusion; Whence we conclude that the image of a straight line 
perpendicular to the axis of the lens which forms it, is 
a conic section, and comparing the two Equations, we 
find, 


A{ l- e*) 

1 -f- e cos v * 


Equations 
compared; 


f" = r, 


!—«■) = 


JT’ 



(52) 


(53) 









ELEMENTS OF OPTICS. 


•225 


For the same lens, F is constant: its value ?. * in ^ ( i uatlon (^) 

’ II ’ ’ sbovra that the 

Equation (52), which is the radius of curvature at the l^Xni be one 
vertex, is also constant. oftheconio 

From Equation (53), it is easily seen that the curve 
will be the arc of a circle, ellipse, parabola, hyperbola, 
or a right line, one of the varieties of the hyperbola, ac- - 
cording as 


f 





Conditions for 
the different 
eonio sections. 




or according as the distance of the object is infinite; 
greater than the principal focal distance of the lens; 
equal to this distance; less than this distance; or zero. 

If the section P Q be supposed to revolve about the 
axis of the lens, it will generate a plane, and the image 
a curved surface whose nature will depend upon the dis¬ 
tance of the object. 

We have seen that a positive value for f'\ answers sign of the focal 
to a virtual, and a negative value to a real focus; “ Lstance /? r * . 
so, if the points of the image be indicated by 'positive whether the 
values for f'\ the image will be virtual; if by nega- 1 “ r ^ 3realor 
tive values, real. For a concave lens, F u is positive, 
and Equation (51), answers to this case. For a convex 
lens, F i{ is negative, and Equation (51), becomes 
15 



226 


NATURAL PHILOSOPHY. 


Image will be 
real for a convex 
lens as long as 
the object is 
beyond the 
principal focus; 


f" 



F 

M a _ 

F 

—— cos 0 


. . . (54) 


and the image will always be real as long as 


cos 6 < 1, 


or 


f 

cos 6 


>F„ 


1.lustration. 


That is, if from the optical centre, with a radius equal 
to the principal focal distance, we describe the arc of a 
circle, and this arc cut the object, the image of all that 
part of the object in¬ 
cluded between the 
points of intersection 
A and A' will be vir¬ 
tual, while that of the 
parts without these lim¬ 
its will be real; if the 
distance of the object 
exceed that of the prin¬ 
cipal focus, the whole 
image will be real. 



§ 63. Multiplying both members of Equation (51), by 
sin 0, it becomes 


Equation (51) 
transformed; 


/".sin 4 = 


F■ f. tan 6 

/ , F 

cos 6 ' •" 


. . ( 55 ) 


and giving to d, its greatest value for any assumed object, 
f ^ an & will be the length of that portion of the object on 








ELEMENTS OF OPTICS. 


227 


the positive side of the axis as long as & is positive and less Explanation of 
than 90° ; f" sin d, is the distance of the extreme limit of terms ’ 
the image of this portion of the object from the axis; and 
writing 


f tan 6 = S, 

f" sin a = 


Substitutions; 


Equation (55) becomes, after dividing both members by 
/ tan 0, 


F„ 


cos 6 


+ F„ 


Equation (55) 
transformed, 


If the linear dimensions of the object be small as com¬ 
pared with its distance from the optical centre, we may compared with 
write unity for cos 0, the image will, § 48, and Eq. (52), its distance from 
sensibly coincide with and the above equation 
reduces to 


t f+K 


. . (56). 


In which the essential signs of all the quantities correspond 
to a concave lens. For a convex lens, F u is negative, and 
Equation (56) becomes 


° it _ JPjj (K7) Equation for a 

§ J _ Ji 1 * * * * ' '* convex lens; 


Equations (51) and (56), show that the image of every 
real object formed by a concave lens is virtual, erect, 
and less than the object, while Equations (54) and (57), 
show that the image of every real object formed by a 
convex lens is real as long as the object is beyond the 

. lii Images formed 

principal focus, is inverted, and less or greater than the by concave and 
object, depending upon the distance of the latter from convex lenses. 







' 228 


NATURAL PHILOSOPHY. 


when the image the optical centre. "When the distance of the object is 
r:;r lt0 tw i° e that of the principal focus, Equation (57) becomes 



other cases. and the object and image are equal in size. When the 
object is within twice the principal focal distance, it is 
less, and when beyond this same distance it is greater 
than the image. 

Relation between § 64. If we make 0 equal to nothing in Equation (51), 
ii nea rd im e nsi°ns j?rr w *q co i nc iq e with the axis of the lens, its length 
imago; will measure the distance of the image from the opti¬ 
cal centre, while f will measure that of the object on 
the same line. Denoting these distances by D u and D , 
respectively, substituting them in Equation (51), clear¬ 
ing the fraction in the second member, and dividing 
both members by 2>, we find 


A, _ 

D f + Fu 

which, in Equation (56), gives 
5 ~ D 


(58) 


same in words. That is to say, the corresponding linear dimensions of an 
object and of its image are to each other directly as 
their respective distances from the optical centre. 

image formed by § 65. If an image be formed by deviation at a sin- 
•higitsurfalc- sur ^ ace ) its points will be readily found by means of 
Equation (36); the optical centre, in this case, being 
at the centre of curvature § 60. 

Writing f for c, and f for c\ that Equation becomes 





ELEMENTS OF OPTICS. 


229 


JL_ 

7 


m—l m 

~ + 7 ; 


Equation 

applicable; 


making y = oo, 


1 _ m—l 1 

7' “ “ “ T? 


• . (59) 


Principal focal 
distance; 


hence, 


or, 


JL 

f~ F, 


+ 



J -f+mF, 


F. 


1 + 


to i?) 

7 T 


Equation for 
discussion; 


Same in another 
form; 


For an oblique pencil passing through the optical cen¬ 
tre, we have, on the supposition that the object is a 
right line perpendicular to the axis of the surface, 


7 = 


F. 


1+ ?^cos* 


/ 


( nr\\ Same for an 
\yV) oblique pencil 
through the 
optical centre. 


wherein - y - = i, as in article (62). 


§ 66. If the image he formed by reflexion, to = 
and Equation (60) becomes 


1, 


/'=- 


F. 


Image formed by 


Tp 

1 -f L. COS < 
J 


. ■ (61) 

reflexion; 












230 


NATURAL PHILOSOPHY. 


ninstrotion; since for a concave re¬ 
flector, F n Eq. (59), 
becomes negative. This 
is a polar equation of a 
conic section, the nature 
of which will result 
from the relation of F t 
to f. It will, § 62, be 
an ellipse, parabola, or 
hyperbola, according as 

Image of a right f > F f \ f = F f \ Or f < F\ 

line will be a 
conic section. 

§ 67. By a process entirely similar to that of § 63 and 

XC6l3*tlOH DObWGCu 

dimensions of §64, we shall find that the linear dimension of the ob- 
object and image. j ec ^. ^ ^ corresponding dimension of the image , as the 

distance of the object from the centre is to that of the 
image from the same point. And a moment’s reflec¬ 
tion will .show us that all real images must be in front, 
while all virtual images must be behind the reflector. 

§ 68. We get the point in which the image cuts the 
axis by making 



Equation for 
discussing a 
concave reflector; 

or 


= o, 


/'=- 




/ 




(62) 


interpretation of This value of f bein S negative, the image will be 
results; found between the reflector and the centre, the distance/ 

being positive on the opposite side. As long as /is posi¬ 
tive, the image will, lie between the centre and reflec¬ 
tor, f will be less than f and the image, consequently, 
less than the object. When / is zero, /' will also equal 
zero, and the object and image will be equal and occupy 









ELEMENTS OF OPTICS. 


231 


the centre. When f becomes negative, or the obiect Posltions and 
passes between the centre and reflector, f will be posi- the image when 
tive as long as/* < F t , and the image will pass without, 
f will be greater than/ 1 , or the image will be greater than centre and the 
the object. When/*, being still negative, is equal to F t vertex - 
or the object is in the principal focus, the image will be 
infinitely distant. The object still approaching the reflec¬ 
tor, f will be greater than F l ; f becomes negative again 
and the image will approach the reflector from behind it, 
and will be greater than the object till f — 2 F t or the 
object be in contact with the reflector, when/ 1 ' will equal 
y, and the image and object be of the same size. 

§ 69. When the reflector is convex, r is negative, the Convex reflector, 
principal focal distance F t Equation (59), is positive, and 
Equation (60) becomes 


/' = 



and making 4 = 0, 


(63) 


Equation 
applicable; 




^04). Equation for 
discussion: 


This valufe of f is always positive, greater than F^ and Eelation3 
less than 2 F ), for all values of/*, between 2 F t and in- between the 
finity, or for any position of the object from the surface ^reaTobje 1 ^ 0 
of the reflector to a point infinitely distant in front. In 
the latter position, f is equal to F t , or the image is in 
the principal focus. It follows also, that the image, which 
will always be virtual for real objects, will be elliptical, 
erect, and smaller than the object. 


§ 70. If we make f positive, greater than F t and less same for viituai 
than 2 the object will be virtual; the image real, objecta * 
erect, and greater than the object. 





2.32 


NATURAL PHILOSOPHY. 


OF THE EYE AND OF VISION. 

The eye; § 71. The eye is a collection of refractive media which 

concentrate the waves of light proceeding from every 
point of an external object, on a tissue of delicate nerves, 
called the retina, there forming an image, from which, 
by some process unknown, our perception of the object 
arises. These media are contained in a globular en- 
Fonr coatings Ve l 0 pe composed of four coatings, two of which, very 
refractive media; unequal in extent, make up the external enclosure of the 
eye, the others serving as lining to the larger of these 
two. 



shape of the eye; The shape of the eye is spherical except immediately 
in front, where it projects beyond the spherical form, 
as indicated at d e d ", which represents a section of the 
human eye through the axis by a horizontal plane. This 
The cornea; p ar t ca p ec [ the cornea , and is in shape a segment of 
an ellipsoid of revolution about its transverse axis which 
coincides with the axis of the eye, and which has to 
the conjugate axis, the ratio of 1,3. It is a strong, 
horny, and delicately transparent coat. 

Immediately behind the cornea, and in contact with 




ELEMENTS OF OPTICS. 


233 


It, is the first refractive medium, called the aqueous Mueons 
humour , which is found to consist of nearly pure wa- humour5 
ter, holding a little muriate of soda and gelatine in 
solution, with a very slight quantity of albumen. Its 
refractive index is found to be very nearly the same as 
that of water, viz. : 1,336, and parallel rays having the 
direction of the axis of the eye will, in consequence of 
the figure of the cornea, after deviation at the surface 
of this humour, converge accurately to a single point. 

At the posterior surface of the chamber A, in con¬ 
tact with the aqueous humour, is the iris, g g, ; 
which is a circular opaque diaphragm, consisting of 
muscular fibres by whose contraction or expansion an 
aperture in the centre, called the pupil, is diminished Pn P u * 
or increased according to the supply of light. The ob¬ 
ject of the pupil seems to be, to moderate the illumi¬ 
nation of the image on the retina. The iris is seen 
through the cornea, and gives the eye its color. 

In a small transparent bag or capsule, immediately 
behind the iris and in contact with it, closing up the 
. pupil, and thereby completing the chamber of the aque¬ 
ous, lies the crystalline humour , B; it is a double con- crystalline 
vex lens of unequal curvature, that of the anterior sur- llumour; 
face being least; its density towards the axis is found 
to be greater than at the edge, which corrects the 
spherical aberration that would otherwise exist; its 
mean refractive index is 1,384, and it contains a much 
greater portion of albumen and gelatine than the other 
humours. 

The posterior chamber 0, of the eye, is filled with 
the vitreous humour , whose composition and specific vitreons 
gravity differ but little from those of the aqueous. Its re- humour 5 
fractive index is 1,339. 

At the final focus for parallel rays deviated by these hu¬ 
mours, and constituting the posterior surface of the cham¬ 
ber C, is the retina , h h h, which is a net-work of nerves 
of exceeding delicacy, all proceeding from one great 0ptIc nem 
branch 0 , called the optic nerve , that enters the eye 



234 


NATURAL PHILOSOPHY. 


Graphic 

representation of 
the eye; 


Fig. 42. 



obliquely on the side of the axis towards the nose. The 
retina lines the whole of the chamber C, as far as i i, 
where the capsule of the crystalline commences, 
choroid coat ; Just behind the retina is the choroid coat , &&, cov¬ 

ered with a very black velvety pigment, upon which 
the nerves of the retina rest. The office of this pig¬ 
ment appears to be to absorb the light which enters the 
eye as soon as it has excited the retina, thus prevent¬ 
ing internal reflexion and consequent confusion of vision, 
sclerotic coat; The next and last in order is the sclerotic coat , which 
is a thick, tough envelope d d'd ", uniting with the cor¬ 
nea at d d’\ and constituting what is called the white 
of the eye. It is to this coating that the muscles are 
attached which give motion to the whole body of the 
eye. 

inverted images From the description of the eye, and what is said in arti- 
retl^ d0nthe c * e (62), it is obvious that inverted images of external 
objects are formed on the retina. This may easily be seen 
by removing the posterior coating of the eye of any re¬ 
cently killed animal and exposing the retina and cho¬ 
roid coating from behind. The distinctness of these im¬ 
ages, and consequently of our perceptions of the objects 
from which they arise, must depend upon the distance 





ELEMENTS OF OPTICS. 


235 


of the retina from the crystalline lens. The habitual Habitual P° sitioa 
position of the retina, in a perfect eye, is nearly at the ° f the retma ’ 
focus for parallel rays deviated by all the humours, be¬ 
cause the diameter of the pupil is so small compared with 
the distance of objects at which we ordinarily look, that 
the rays constituting each of the pencils employed in 
the formation of the internal images may be regarded 
as parallel. But we see objects distinctly at the distance 
of a few inches, and as the focal length of a system of 
lenses, such as those of the eye, Equation (25), increases 
with the diminution of the distance of the radiant or 
object, it is certain that the eye must possess the power Eye possesseg tbe 
of self-adjustment, by which either the retina may be power of 
made to recede from the crystalline humour and the self ' adju9tment ‘ 
eye lengthen in the direction of the axis, or the curva¬ 
ture of the lenses themselves altered so as to give greater 
convergeney to the rays. The precise mode of this ad¬ 
justment does not seem to be understood. There is a 
limit, however, with regard to distance, within which 
vision becomes indistinct; this limit is usually set down 
at six inches , though it varies with different eyes. The Limit of distinct 
limit here referred to is an immediate consequence of vision; 
the relation between the focal distances expressed in 
Equation (25), for when the radiant or objectis brought 
within a few inches, the corresponding conjugate or im¬ 
age is thrown behind the point to which the retina may 
be brought by the adjusting power of the eye. 

With age the cornea loses a portion of its convexity, 
the power of the eye is, in consequence, diminished, and 
distinct images are no longer formed on the retina, the Long si » htedr,eal 

° ° . . . and its remedy; 

rays tending to a focus behind it. Persons possessing 
such eyes are said to be long sighted , because they see 
objects better at a distance; and this defect is remedied 
by convex glasses, which restore the lost power, and with 
it, distinct vision. 

The opposite defect arising from too great convexity 
in the cornea is also very common, particularly in young 
persons. The power of the eye being too great, the 




236 


NATURAL PHILOSOPHY. 


Shortsightedness 
and its remedy. 


Images on the 
retina are 
inverted; 


Bat objects 
appear erect; 


Explanation of 
the above. 


Base of the optic 
nerve insensible 
to light 


image is formed in the vitreous humour in front of the 
retina, and the remedy is in the use of concave glasses. 
Cases are said to have occurred, however, in which the 
prominence of the cornea was so great as to render the 
convenient application of this remedy impossible, and 
relief was found in the removal of the crystalline lens, 
a process common in cases of cataract, where the crys¬ 
talline loses its transparency and obstructs the free pas¬ 
sage of light to the retina. 

The fact that inverted images are formed upon the 
retina, and we, nevertheless, see objects erect, has given 
rise to a good deal of discussion. "Without attempting 
to go behind the retina to ascertain what passes there, 
it is believed that the solution of the difficulty is found 
in this simple statement, viz.: that we look at the object , 
not at the image. This supposes that every point in an 
image on the retina, produces, without reference to its 
neighboring points, the sensation of the existence of the 
corresponding point in the object, the position of which 
the mind locates somewhere in the axis of the pencil 
of rays of which this point is the vertex; all the axes 
cross at the optical centre of the eye, which is just within 
the pupil, and although the lowest point of an object 
will, in consequence, agitate by its waves the highest 
point of the retina affected, and the highest point of 
the object the lowest of the retina, yet the sensations be¬ 
ing referred back along the axes, the points will appear 
in their true positions and the object to which they 
belong erect. In short, instead of the mind contemplat¬ 
ing the relative positions of the points in the image, the 
image is the exciting cause that brings the mind to the 
contemplation of the points in the object. 

It may be proper to remark here, that the base of 
the optic nerve, where it enters the eye, is totally insen¬ 
sible to the stimulus of light, and the reason assigned for 
this is, that at this point the nerve is not yet divided 
into those very minute fibres which are capable of being 
affected by this delicate agent. 



ELEMENTS OF OPTICS. 


287 


§ 72. All other things being equal, the apparent magnitude Apparent magni 
of an object is determined by the extent of retina covered S^mined? bje °' 
by its image. 


Fif. 48. 



If, therefore, be a section of the retina, by a 

plane through the optical centre (7, of the eye, and 
A B—l, a b = X, the linear dimensions of an object and 
its image in the same plane, we shall have, from the 
similar triangles GAB and C a 5, 



(65) 


Dimension of 
image of an 
object on the 
retina; 


denoting by e, the distance of the object. And for any 
other object whose linear dimension is V and distance s /? 
calling the corresponding dimension of the image X, 


\ = - 



Same for a second 
object; 


and since G a is constant, or very nearly so, 


* S S 

J 


Proportion 


that is, the apparent linear dimensions of objects are as 
their real dimensions directly , and distances from the 

eye inversely. But A, may betaken as the measure of Kule fiKt ’ 
the angle B C A = b G a, which is called the visual an- 









238 


NATURAL PHILOSOPHY. 


Eoie second, gle, and hence the apparent linear magnitudes of objects 
are said to be directly proportional to their visual angles. 

Small and large objects may, therefore, be made to ap¬ 
pear of equal dimensions by a proper adjustment of their 
distances from the eye. For example, if X = X y , we have 

Example for 
illustration, 

or, 



Numerical data; and if 1= 1000 feet,.s = 20000, and V = 0,1 of a foot, 
or little more than an inch, 

RsSQlt 6 _ 20000 . 0,1 _ t 

' 1000 7 ’ 

the distance of the small object at which its apparent 
magnitude will be as great as that of an object ten thou¬ 
sand times larger, at the distance of 20000 feet. 



MICROSCOPES AND TELESCOPES. 

Microscopes; g 73. From what has just been said, it would appear 
that there is no limit beyond which an object may not be 
magnified by diminishing its distance from the optical 
centre of the eye. But when an object passes within the 
limit of distinct vision, what is gained in its apparent in- 
Expianatory crease of size, is lost in the confusion with which it is seen, 

remarks; If, however, • while the object is too near to be distinctly 

visible, some refractive medium be interposed to assist the 
eye in bending the rays to foci upon its retina, distinct 
vision will be restored, and the magnifying process mav 





V 


_ L /O-i 

JdA Slunv^p 


ELEMENTS OF OPTICS. 

Jll S/u'nx<f 


239 


be continued. Such a medium is called a single micros- single 
cope, and usually consists of a lens, whose principal focal microscoi>e: 
distance is negative and numerically less than the limit of 
distinct vision. 

To illustrate 
the operation of 
this instrument, 
let MJV be a 
section of a dou¬ 
ble convex lens 
whose optical 
centre is C; 

Q P an object in front and at a distance from C equal 
to the principal focal distance of the lens; E the optical 
centre of the eye, at any distance behind the lens. 

The rays Q C and P C, containing the optical centre, 
will undergo no deviation, and all the rays proceeding 
from the points Q and P, will be respectively parallel to 
these rays after passing the lens; some rays, as JST i? Explana(ion of 
from Q, and ME from P, will pass through the optical the figure; 
centre of the eye, and will belong to two beams of light 
whose boundaries will be determined by the pupil, and 
whose foci will be at q and jp on the retina, giving the 
visual angle, 


Fig. 44. 



Its operation 
illustrated: 


MEN = PC Q: Ec,at,on 

7 same; 

or the apparent magnitude of the object P Q, the same 
as if the optical centre of the eye were at that of the 
lens. And this will always be the case when an object 
occupies the principal focus of a lens whatever the dis¬ 
tance of the eye, provided the latter be within the field 
of the rays. 

Without tlie lens, the visual angle is QEP <P CQ ; Effect of the 
hence, the apparent magnitude of the object will be in- slngle 

1 x ° ° microscope, 

creased by the lens. 

Calling X and X, the apparent magnitudes of the ob¬ 
ject as seen with, and without the lens, we shall have, 







240 


NATURAL PHILOSOPHY. 


Magnitudes of 
au object with 
and without the 
lens compared; 


. . PQ.PQ.. J_.J_ 

: GQ ' C Q' EQ' 


or, by using the notation employed in Equation (33), and 
calling E Q , the limit of distinct vision, unity, 


when the iens As long as F u < 1, or the principal focal length of 
may be used as a ^ i eng - g j egg ii m it of distinct vision, the ap- 

microscope; parent size of the object will be increased, and the lens 
may be used as a single microscope. 

We can now understand what is meant by the power 
of a lens or combination of lenses, referred to at the close 

What is meant 

by the of article (39). —-, which was there said to measure 

magnifying n 

power of a single t p e p 0wer 0 f a l e ns, we see f rom Equation (66), expresses 

microscope; x \ x N A 

the apparent magnitude of an object compared to that 
at the limit of distinct vision, taken as unity; and what¬ 
ever has been demonstrated of the powers of lenses gen¬ 
erally, is true of magnifying powers. Thus, in Equation 
(31), we have the magnifying power of any combination 
of lenses equal to the algebraic sum of the magnifying 
powers taken separately. Should any of the individuals 
of the combination be concave, they will enter with signs 
contrary to those of the opposite curvature. 

Rule for Th e power °f a single microscope is , Equation (66), 

power of a°singio to the limit of distinct vision divided by its princir 

pricroscoie; pal focal distance , and the numerical value of the power 

will be greater as the refractive index and curvature are 
greater. 


§ 74. To obtain a general expression for the visual an¬ 
gle under which the image of an object formed by a 
lens, and having any position in reference to the eve, 





ELEMENTS OF OPTICS. 


is seen, let Q P , be 
an object in front of 
a concave lens. From 
P, draw through the 
optical centre P, the 
line P Ej from P, 



Fig. 45. 


To find tho 
visual angle 
under which an 
image formed by 
a lens is seen; 


draw the extreme 


ray P M, and from 

M draw M S , making with P M produced the angle SMT 
equal to the power of the lens; then will, § 47, MS be 
the corresponding deviated ray, and its intersection y?, 
with the ray P P, through the optical centre, will be a 
point in the image; from y>, draw jp q, parallel to P Q, 
till it is cut by the ray Q E, through the other extreme 
of the object and optical centre; jp q will be the image. 
Let 0 , be the optical centre of the eye; then denoting 
the visual angle jp 0 q by A, we have, 


Value of visual 
angle; 


A= M. = _ 


Oq Eq-OE 


and representing the distances Q E by /, E q by /", 
and E 0 by d, we find, 


2 I> = Q-P.y--, Eq-EO=f" -d\ 


and hence 



Same in other 
terms; 


and denoting the visual angle PEQ by A', 


A /" 1 


Batio of visual 


A'~ f"-d~ , ± ' 


r 


1 /" 


(07) angles with and • 
without the lens. 


16 













242 


NATURAL PHILOSOPHY. 


\ 


Sign of this 
ratio depends 
upon; 


Eye placed so as 
to see the image 
formed by a 
eoncave lens; 


The angles A and A! will have contrary signs when on 
opposite sides of the axis of the deviating surface. 
The relation expressed by this equation answers to a 
concave lens in which f" will, Equation (27), be posi¬ 
tive for a real object. Moreover, d is positive, the eye 


being on the same 
side of the lens as 
the object; but that 
the image may be 

seen the eye must - 

be on the opposite 
side, in which case 
d will be negative, 
and the Equation 
becomes 


Fig. 46. 



* t.L • > 


Equation 
corresponding to 
this case; 


A' 


1 + 


dj 

f" 


( 68 ) 


Eeal image 
formed by a 
convex lens; 


whence we conclude that objects will always appear dimin¬ 
ished when seen through concave lenses. 

If the lens be 
convex and the 
object be situa¬ 
ted beyond its 
principal focus 
f” will be nega¬ 
tive, and Equa¬ 
tion (68) becomes 



Equation 
corresponding; 



Distinct vision 

supposed possible anc [ supposing distinct vision possible for all positions 

for all positions „ ., . A x 

of the eye. ot the e ye, it. appears, 











ELEMENTS OF OPTICS. 


243 


1st. That when the object is at a distance from the Conclusion Ami 
lens greater than that of the principal focus, in which 
case there will be a real image, the lens will make 
no difference in the apparent magnitude of the object, 
provided the eye is situated at a distance from the 
lens equal to twice that of the image. 

2d. At all positions for the eye between this limit second, 
and the image, the apparent magnitude of the object 
is increased by the lens. 

3d. At a position half way between this limit and Third, 
the lens, the apparent magnitude of the object would 
be infinite. 

4th. The eye being placed at a distance greater than Fourth . 
twice that of the image, the apparent magnitude of the 
object will be diminished by the lens. 

5th. When the distance of the object from the lens F . fth 
is equal to that of the principal focus, in which case /" 
becomes infinite, the apparent magnitude will be the 
same as though the eye were situated at the optical centre 
of the lens, no matter what its actual distance behind 
the lens. . 4 

§ 75. The visual angle under which the image formed to find the visnkt 
by a reflector is seen, is found in the same way. Thus, let ® 1nnd ® r 

P Q be an object in 
front of a convex re¬ 
flector M AT. From 
the extreme point P, 
of the object, and 
through the optical 
centre C, draw the 
ray P C\ from the 
same point P, draw to 
the extrema of the 

reflector the ray P M, and from M draw MS, making 
with P M, the angle P MS equal to the power of the Explanation; 
reflector; MS will, § 53, be the deviated ray, and its 
intersection with P C, will give the image of the point 


Fig. 4a 
A 

t 


formed hy a 
reflector Is Been; 






244 


NATURAL PHILOSOPHY. 


Construction of 
the image formed 
by a reflector; 


P. Draw p q, paral¬ 
lel to P Q , till it is 
intersected by Q C\ 
drawn through the op¬ 
posite extreme of the 
object and optical 
centre, and we have 
the image. Let the 
optical centre of the 
eye be at 0 ; then, de¬ 
denoting the visual angle p 0 q by A, will 



Value of visual 
angle with the 
reflector; 


A = 


Ll = QJ^ C 2- 

Oq G Q Oq 9 


and representing, as before, C Q, Cq, and C 0 , by f, 

P O 

f\ and d , respectively, and the visual angle _^by A\ 

C Q 


we have 


Ratio of visual 
angles with and 
without the 
reflector. 

We shall not stop to discuss this Equation. 

in practice §76. ¥e have supposed, in the preceding discussion, 
distinct vision is distinct vision to be possible for all positions of the eye : 

not possible for r r «/> 

aii positions of but this we know depends upon the state of convergence 
toe eye; or divergence of the rays. If, however, the image, 
when ‘one is formed, instead of being seen by the naked 
eye, be viewed by the aid of another lens, so placed 
that the rays composing each pencil proceeding from 
the object shall, after the second deviation, be parallel, 
or within such limits of vergency that the eye can 
command them, the object will always be seen distinctly, 
And the image is and either larger or smaller than it would appear to 
therefore viewed ^ unlisted eye, depending upon the magnitude of 
lens. the image, and the power of the lens used to view it. 


A 

A' 


_ /' 
~ d-r 


(TO) 








ELEMENTS OF OPTICS. 


245 


As most eyes see distinctly with plane waves or parallel Position of the 
rays, this second lens is usually so placed that the image eye 1<m 
shall occupy its principal focus; and where this is the 
case, we have seen that the apparent magnitude of the 
image will be the same as though the eye were at its 
optical centre. 



Fig. 49. 


Refracting 

telescope; 


The image p q, in the principal focus 

of the lens m n, draw from the point y?, the line Construction for 
p 0, to the optical centre of this lens; the rays from p ughuraldTo 
will, § 73, be deviated parallel to this line, and the line the retina - 
O' K, through the optical centre O' of the eye, paral- 
rel to p 0, will determine by its intersection K, 
with the retina, the place upon that membrane of the 
image of the point P. 

Calling the principal focal distance of this lens, (F„) ; 
in Equation (67), will equal f" + and that equa¬ 
tion will become, by first making f" and d negative and 
then replacing d by this value, 


a _ r_ 

3’~ (K) 


General equation 
(71) made applicable 
to this telescopo; 


and if the object P Q, be so distant that the rays com¬ 
posing each of the small pencils whose common base is 
M JV, may be regarded as parallel, f" becomes F u , 
and we have, 


A' ~~ 



Ratio of visual 
( 72 ) angles for parallel 
rays; 










216 


NATURAL PHILOSOPHY. 


Fig. 49. 



Compound 
microscope; 

Field and eye 
lenses; 


Knlo for 

magnifying 

power. 


Otgccts appear 
inverted. 


Galilean 
telescope; 


Construction of 
image on the 
retina; 


Equation (71) exhibits the principles of the com- 
pound refracting microscope, and refracting telescope; and 
Equation (72), which is a particular case of (71), those 
of the astronomical refracting telescope. The lens MN, 
next the object, is called the object or field lens, and m n, 
the eye lens. The magnifying power in the first case, is 
equal to the distance of the image from the field lens 
divided by the principal focal length of the eye lens / 
and in the second, to the principal focal length of the 
field lens , divided by that of the eye lens. 

The ratio of A to A', being negative, shows that ob¬ 
jects appear inverted through these instruments, the vis¬ 
ual angles of corresponding parts of the object and im¬ 
age being on opposite sides of the axis. 

§ 77. If instead of a convex, a concave lens be used 
for the eye lens, the combination will be of the form used 
by Galileo, who invented this instrument in 1609. In 
this construction, the eye lens is placed in front of the 
image at a distance equal to that of its principal focus, so 
that the rays composing each pencil shall emerge from it 
parallel. 

Draw through the point p , where the image of P 
would be formed, the line p 0 , to the optical centre O of 
the eye lens, and through the optical centre O’ of the eye, 
the line O' ^parallel top 0 , its intersection K, with the 
retina will give the image of the point P on the back 
part of the eye. 








ELEMENTS OF OPTICS. 


247 



The rule for finding the magnifying power of this Magnifying 
instrument is the same as in the former case; for we power found 

7 analytically; 

have, 

d=f" -(F,)-, 


which in Equation (67), after making/", and d, negative, 
gives 



(73) Ratio of visual 

angles; 


and for parallel rays, 

A ~ {F u ) ' 


(74) 


Same for parallel 
rays; 


The second member being positive, shows that -objects objects appear 
seen through the Galilean telescope appear erect. 


§ 78. If we divide both numerator and denominator of 
Equation (72), by F „. (F„), it becomes, 


1 


1 ’ 

Magnifying 

power in terms 
of the powers of 

F„ 

the lenses; 



and denoting by L , the power of the field, and by Z, that 
of the eye lens, we have 


A___ _ ± 

A' ~ L 


(75) Ratio of visual 
angles; 












248 


NATURAL PHILOSOPHY. 


Rule for 

magnifying 

power. 


that is, the magnifying power of the astronomical tele¬ 
scope is equal to the quotient arising from dividing the 
j power of the eye lens Inj that of the field lens. 



General 
explanation; 


Field of view; 


Determined by 
construction; 


§ 79. If E\ be the optical centre of the field, and O 
that of the eye lens of an astronomical telescope, the 
line E 0 , passing through the points E and O , is called 
the axis of the instrument. Let Q r P r be any object 
whose centre is in this axis, and q' p' its image. How, 
in order that all points in the object may appear equally 
bright, it is obvious from the figure, that the lens must 
be large enough to embrace as many rays from the points 
P’ and as from the intermediate points. It is not 
so in the figure; a portion, if not all the rays from those 
points will be excluded from the eye, and the object, in 
consequence, appear less luminous about the exterior 
than towards the centre, the brightness increasing to a 
certain boundary, within which all points will appear 
equally bright. The angle subtended at the centre of the 
field lens, by the greatest line that can be drawn within 
this boundary, is called the field of view. To find this 
angle, draw m N and Mn to the opposite extremes of the 
lenses, intersecting the image mp and and the axis in 
X ; then will p q be the extent of the image of which 
all the parts will appear equally bright. Draw qEQ 
andy? E P; the angle PE Q —p Eq, is the field of view, 
which will be denoted by l ; 



First form of its 
value; 


(76) 







ELEMENTS OF OPTICS. 


249 


but 


rm r, _ Transformations; 

pq = ^L.Xr .( 77 ) 

to find X 0 and X r, call the diameter MX of the object 
lens a, that of the eye lens /3, and we have 

a: (3 :: EX: X0 proportions,- 

* + (3:(3::XX+XO:XO 


hence, 


XO=.JL.(/" + (2Q); 

a+ P 

and in the same manner, 


ZX=«.(f"+(F ll )); 

a+ (3 


Eelations from 
the figure; 


Xr=f" —EX =f"— *(J" + (F, t )) 

a+p 


« + £ ’ 


these values in Equation (77), give 


jpq = 


r+m ’ 


Substitutions; 


and this in Equation (76), gives, by introducing the 
powers of the lenses, 


y _ T (3 l “ a Ij 

* l + L 


/hq\ Final value for 
\‘°) field of view. 


The rays of each of the several pencils emerging from 
the eye lens parallel, will be in condition to afford dis- 










250 


NATURAL PHILOSOPHY. 


Fig. 51. 



Proper position' 
for the eye 
indicated in 
telescopes; 


tinct vision, and the extreme rays m O' and n O', will 
be received by an eye whose optical centre is situated 
at O'. If the eye be at a greater or less distance than 
O', from the eye lens, these rays will be excluded, and 
the field of view will be contracted by an improper posi¬ 
tion of the eye. It is on this account that the tube con¬ 
taining the eye lens of a telescope usually projects a 
short distance behind to indicate the proper position for 
the eye. 

From the similar triangles p 0 q and m O' n, we have 


Distance of 
optical centre of 
the eye from that 
of the eye lens; 


0 0' = rO = f 7 (J + ^ . (F„) .... (79) 

pq pl — aL 


Position of the This also applies to the Galilean instrument, by chang- 
GaUhL the i n g the sign of l, which will render 0 O', negative, 

telescope; The eye should, therefore, be in front of the eye-glass 

in order that it may not, by its position, diminish the 
field of view; but as this is impossible, the closer it is 
placed to the eye-glass the better. 

Arrangement for When the telescope is directed to objects at different 
dtocTbetween distances, the position of the image, Equation (27), will 
the lenses. vary, and the distance between the lenses must also be 
changed. This is accomplished by means of two tubes 
which move freely one within the other, the larger usu¬ 
ally supporting the object and the smaller the eye lens. 


Terrestrial 
telescope ; 


§ 80. The terrestrial telescope is a common astronomi¬ 
cal telescope with the addition of what is termed an 








ELEMENTS OF OPTICS. 


251 


erecting piece , which consists of a tube supporting at Erecting 
each end a convex lens. The length of this piece should 
be such as to preserve entire the field of view, and its 
position so adjusted that the image formed by the object 
glass, shall occupy the principal focus of the first lens 
of the erecting piece, as indicated in the figure, 



Terrestrial 
telescope „ 


in which case a second image will be formed in the prin¬ 
cipal focus of the second lens of the erecting piece, and 
the corresponding linear dimensions of these images will 
be to each other as their distances from the lenses whose Belati(>nbetween 
principal foci they occupy, Equation (72). These images the two images 
being viewed through the same eye lens, viz.: that 0 f formed ’ 
the telescope, their apparent, will be directly as their real 
magnitudes. Hence, denoting by A and A" the visual 
angles subtended at the optical centre of the eye lens by 
the first and second images respectively; by l' and l" the 
powers of the first and second lenses of the erecting 
piece, we have, 


Jpt jp y Ratio of their 

- —_ tiL — _• visual angles at 

A F ej l" * the optical centra 

of the eye lens; 

in which F et and F tu , are the principal focal lengths of 
the first and second lenses. Multiplying this by Equation 
(75), member by member, we have for the magnifying 
power of the terrestrial telescope, 

Magnifying 

W power of the 

_ - jf • terrestrial 

A! L Z telescope. 


A" 







252 


NATURAL PHILOSOPHY. 


objects appear And since the ratio of A" to A' is positive, objects will 
appear through this instrument erect. 


S3 U 0 nd § 81. If, now, the object approach the field lens, 

in Equation (71), will increase, and the magnifying power 
become proportionably greater; but this would require 
the tube containing the eye lens to be drawn out to ob¬ 
tain distinct vision, and to an extent much beyond the 
limits of convenience if the object were very near. This 
difficulty is avoided by increasing the power of the ob¬ 
ject lens, as is obvious from Equation (54); and when 
this is carried to the extent required for very great prox¬ 
imity, the instrument becomes a compound microscope , 
which is employed to examine minute objects. The corn- 


rig. sa 


Compound 
microscope; 



same in principle pound microscope not differing in principle from the 
tei3pe? S telescope, its magnifying power is given by the Equa¬ 
tion (68.) 

Its magnifying 
power; 



and substituting for its value in Equation ( 40 ), we 

J 

have for a convex object lens 


A 
A' 


1 


{FA 


’ 1 


1 


T’ 


Same in a 
different form; 








ELEMENTS OF OPTICS. 


253 


or, writing D for jL ; and representing, as before, the 

J 

powers of the field and eye lenses by L and l , 


A_ 

A' 


D-L’ 


Final value for 

magnifying 

power; 


from which it is obvious that the magnifying power may 
be varied to any extent by properly regulating the po-May be varied; 
sition of the object; but a change in the position of the 
object would require a change in the position of the eye¬ 
glass, and two adjustments would, therefore, be neces¬ 
sary, which would be inconvenient. For this reason, it 
is usual to leave the distance between the lenses unal¬ 
tered and to vary only the distance of the object to suit 
distinct vision. It is, however, convenient to have the 
power of changing the distance between the glasses, as 
by that a choice of magnifying powers between certain 
limits may be obtained, and for this purpose the object g,^“ deye 
and eye glasses are set in different tubes. 


Usual practice; 


different tubes. 


Fig. 54. 


M 



Eeflecting 
telescope; 


§ 82. If the field lens of the astronomical telescope be 
replaced by a field reflector M N, whose optical centre 
is at <7, as indicated in the figure, we have the common 
astronomical reflecting telescope . G being the optical 

centre, d becomes equal to f — and Equation (70), 
becomes, by first changing the sign of f\ and then sub- Explanation; 
stituting this value for 6?, 






254 


NATURAL PHILOSOPHY. 


Magnifying 
power for 
terrestrial 
objects; 


Same for 
celestial objects. 


To obviate the 
interception of 
light by the 
observer’s head; 


Herschel’8 

Instrument 

» 


Gregorian 
telescope; 


A __ iL 

* ~ (A,) 

and for plane waves or parallel rays, 

A _ _A_ 
a (A,) ' 

hence, the rule for the magnifying power is the same 
as for the refracting telescope. 

The figure represents a reflecting telescope of the sim¬ 
plest construction, and it is obvious that the head of the 
observer would intercept the whole of the incident light, 
if the reflector were small, and a considerable portion 
even in the case of a large one; to obviate this, it is 
usual to turn the axis a little obliquely, so that the image 
may be thrown to one side, where it may be viewed 
without any appreciable loss of light. By this arrange 
ment, the image would, of course, be distorted, but in 
very large instruments, employed to view faint and very 
distant objects, it is not sufficient to cause much if any 
inconvenience. This is ITerschel’s instrument. 


§ 83. The* obstruction of light is in a great measure 
avoided in the Gregorian telescope, of which an idea may 
be formed from the figure. 


Fig. 55. 



M N is & concave spherical reflector, of which the op¬ 
tical centre is at (7, and having a circular aperture G H\ an 



















ELEMENTS OF OPTICS. 


255 


image pq of any distant object P Q, is formed by it as Second ima & e 
before; the rays from the image are received by a second Im^concave 
concave spherical reflector, much smaller than the first, reflector ; 
and whose optical centre is at C \ a second image q ', 
is formed by this small reflector, in or near the aper¬ 
ture of the large reflector and is there viewed 
through the eye lens m n. The distance of the position of the 
small reflector from the first image must be greater than 8mallrcflector; 
its principal focal distance, and so regulated that the se¬ 
cond image will be thrown in front of the eye lens, and in 
its principal focus. In order to regulate this distance, the 
small reflector is supported by a rod that passes through a 
longitudinal slit in the tube of the instrument, the rod Device for 
being connected with a screw, as represented in the figure, regulatiDg ' L 
by means of which a motion in the direction of the axis 
may be communicated to it. 

The apparent magnitude of the images p q and p' q\ 
as seen through the same eye-glass at the distance of its 
principal focus, are as their real magnitudes ; and the lat- Relation between 
ter are as the distances of these images from the centre of the two images 
the small reflector, § 67. But by Equation (36), making 
m — — 1, and recollecting that in the case before us, c 
is negative, we have, calling F a , the principal focal dis¬ 
tance of the second reflector, 



Same expreseed 


analytically; 


whence 


c' - F * C 
~ F s -c 


dividing by — c , 


4 


c 


F a A" 
F 2 -c~~A~ 


Same in other 
terms; 





256 


NATURAL PHILOSOPHY. 


Magnifying 
power of 
Gregorian 
telescope; 

May be varied. 


Objects appear 
erect 


Cassegrainian 
telescope; 


Graphic 
representation; 


Magnifying 
power; 


Objects appear 
inverted. 


which, being multiplied by Equation (80), member by 
member, gives for the magnifying power of this tele¬ 
scope, 


Ay_ _ F t f 2 
A' ~ (FJ F 2 — g # 


• • ( 81 ) 


whence, this ratio being positive, the object will appear 
erect; its apparent magnitude may be made as great 
as we please by giving a motion to the small reflector 
which shall cause its principal focus to approach the 
first image, and drawing out, at the same time, the eye 
lens to keep the second image in its principal focus. 


§ 84. If the small reflector be made convex instead 
of concave, we have the modification proposed by M. 
Cassegrain, and called the Cassegrainian telescope , which 
is represented in the figure. Its magnifying power is 


Fig. 56. 



given by Equation (81), by changing the signs of F a 
and 0 , which will give, 

A" F F 

kr ■ * * * * 

and because the ratio of A" to A! is negative, objects 
seen through this telescope with ordinary eye pieces, 
appear inverted. 

















ELEMENTS OF OPTICS. 


257 


§ 85. Sir Isaac Newton substituted for the small cur- Newtonian 
ved reflector a plane one inclined under an angle of 45° telesc<) ^ * 
to tbe axis of tlie instrument, and so placed as to inter¬ 
cept the rays before the image is formed. The vergency 
not being affected by reflexion at plane surfaces, § 55, 
the image is formed on one side, and viewed through 
the lens supported by a small tube inserted in the side of 


Fig. 57. 



Magnifying 

the main tube of the telescope. The magnifying power power found; 
of the Newtonian telescope is given by Equation (81) 
or (82), by making F 2 infinite, in which case <?, becomes 
equal to 2i^ 2 , the distance of the first image from the 
optical centre of the small reflector being sensibly equal 
to the radius of curvature. This gives 


Its value, 



DYNAMETER. 


Dynameter 


§ 86. If any telescope, properly adjusted to view dis¬ 
tant objects, be directed towards the heavens, the field 
lens may be regarded as a luminous object whose im- Digtance ^ 
age will be formed by the eye lens. The distance of object; 

17 













258 


NATURAL PHILOSOPHY. 


the object in this case will be the sum of the princi¬ 
pal focal distances or F u + {Ff, and this being sub¬ 
stituted for /, in Equation (57), we get, by inverting 
and reducing, 


Relation between 
object and image; 



. . (84) 


hence, any linear dimension of the object glass of a tele - 
scope, divided by the corresponding linear dimension of 
Rule. its image , as formed by the eye glass , is egual to the 

magnifying power of the telescope. This is the princi- 
ciple of the Dynameter , a beautiful little instrument 
used to measure the magnifying power of telescopes. 


Illustration 


Construction of 
the dynameter 
explained, 


Fig. 58. 



To understand its construction, let us suppose two lu¬ 
minous circular disks, a tenth of an inch in diameter, 
to be placed one exactly over the other in the principal 
one disk focus m of a lens E, and with their planes at right an- 

eupposed to be ; ° 

moved tangent to gles to its axis ; an image of the common centre of the 
ibe other; disks will be formed on the retina of an eye viewing 
them through the lens, at m". If one of the disks be 
moved to the position m\ so that its 'circumference be 
tangent to that of the other, the image of its centre 
will be at m determined by drawing from O , the. 
optical centre of the eye, a line parallel to that joining 








ELEMENTS OF OPTICS. 


259 


the optical centre of the lens and the centre of the Take onG 
movable disk, article (73); the images will be tangent ^ divkiTd^ 0 
to each other, and the movable disk will have passed 
over a distance equal to its diameter, viz.: one tenth 
of an inch. We now take but one disk, and suppose the 
lens divided into two equal parts by a plane passing 
through its axis; as long as the semi-lenses occupy a 
position wherein they constitute a single lens, an image 
of the disk will be formed as before at m "; but when 
one of the semi-lenses is brought into the position denoted Mov0 one hal L 
by the dotted lines in the figure, having its optical cen¬ 
tre at E\ in a line through m, parallel to m! E, two 
images, tangent to each other, will again be formed; 
for, all the rays from the centre of the disk, refracted 
by the semi-lens in this second position, will be paral¬ 
lel to m E\ and 0 m'", is one of these rays. It is 
obvious also, that the distance E E\ through which the 
movable semi-lens has passed, is equal to the diameter 
of the disk. 

The dynameter 
consists of two tubes 
A B , and C D, mov¬ 
ing freely one with¬ 
in the other, the lar¬ 
ger having a metal¬ 
lic base with an 
aperture in the cen¬ 
tre, over which, to 
qualify the light, is 
placed a thin slip 
of mother-of-pearl 
P. In the oppo¬ 
site end of the 
smaller tube, two 
semi-lenses E E\ 
are made to move 
by each other by 
means of an ar- 


Fig. 



Essential parts of 
the dynameter: 


Fig. 60. 



Arrangement for 
moving the 
serai-lenses; 



































260 


NATURAL PHILOSOPHY. 


Illustration; 


Graduated 
screw head; 


Spring; 


Value of one 
entire turn of 
screw head; 


Value of one 
division. 


Method of using 
the dynameter; 


rangement indicated in the figure, wherein n is a right- 
handed screw with, say, fifty threads to an inch; n' iff 
a left-handed screw, with the same number of threads, 
which works in the 
former about a com¬ 
mon axis, and is 
fastened to the 
frame that carries 
the semi-lens E. 

The screw w, is ren¬ 
dered stationary as 
regards longitudinal motion, by a shoulder that turns 
freely within the top of the frame /S' T at r, and works 
in a nut at V, connected with a frame that carries the 
semi-lens E '; this screw is provided with a large cir¬ 
cular head X X, graduated into one hundred equal 
parts, which may be read by means of an index at X 
or Y, on the frame of the instrument. At t, is a spring 
that serves to press the frames against their respective 
screws, to prevent loss of motion when a change of 
direction in turning takes place. 

When the graduated head is turned once round to the 
right, the semi-lens E\ is drawn up of an inch, while 
the semi-lens E\ is thrust in an opposite direction through 
the same distance, making in all a separation of the opti¬ 
cal centres of ^ of an inch, and the semi-lenses are kept 
symmetrical with regard to the centre of the instrument. 
If the screw had been turned through but one division on 
the head, the separation would have been of ^ 0 r 
nVo of an inch. 

To use the instrument, direct the telescope, whose power 
is to be measured, to some distant object, as a star, and 
adjust it to distinct vision; turn it off the object, and apply 
the dynameter with the pearl end next the eye lens, and 
an image of the object lens will be seen; turn the grad¬ 
uated head, supposed to stand at zero, till two images ap¬ 
pear and become tangent to each other; read the number 
of divisions passed over, and multiply it by gxVo? the pro- 

















ELEMENTS OF OPTICS. 


261 


duct will give the diameter of the image in inches. Mea- MagnIfyiog 
sure by an accurate scale, the diameter of the visible por- pow«r of 
tion of the object glass, which being divided by the mea- telescope foun ^ 
sure of its image just found, will give the magnifying 
power. The index will indicate zero, if the dynameter be 
properly adjusted, when the semi-lenses have their opti¬ 
cal centres coincident. This little instrument is the more 
valuable, because it gives, by an easy process, the magni¬ 
fying power of any telescope, however complicated. 


CAMERA LUCIDA. 

§ 87. This little instrument, the invention of Dr. Wol- C ameraIuci(Ia , 
laston, is of great assistance in drawing from nature. 

In its simplest form, it consists of a glass prism, a section 
of which is represent¬ 
ed by A B C D, with 
one right angle at A, 
and the opposite angle 
^7,135°. Rays proceed¬ 
ing from a point of any 
object S) in front of the 
face A D , enter this 
face without undergo¬ 
ing any material devi¬ 
ation, and being re¬ 
ceived in succession 
by the faces D C and 
CB within the limits of 
total reflexion, they are 
reflected, and finally leave the face B A , in nearly the same 
state of divergence as when they left the object S. The 
eye E\ being so placed that the edge B of the prism Explanation; 
shall bisect the pupil, will receive these rays and bring 
them to a focus r , on the retina, at the same time that 









262 


NATURAL PHILOSOPHY. 


Action of the ^ receive through the half of the pupil not covered 
camera lucida in by the prism, rays proceeding from the point P, of a 
forming images, p enc *^ placed below on a sheet of paper, and bring them 

also to the same focus so that the point in the ob¬ 
ject and point of the pencil will appear to coincide on 


Linear 

dimensions of 
object and image. 


the paper, the whole 
of which will be seen 
through the uncovered 
half of the pupil, and 
a picture of the object 
may thus be traced by 
bringing the pencil 
in succession in ap¬ 
parent contact with 
its various parts. 

The linear dimen¬ 
sions of the picture 
will be to those of the 


Fig. 61. 



\ — 
V 

\ % 

& \ 

\ 

n 

* A 

jS 

7 \ 

\ 


r» 


L' 


JCH 


-IS 


concave lens 
with the camera 
lncida: 


object, as the distance of the camera from the paper, 
to its distance from the object, nearly, 
use of convex or If the paper be very near, the eye may not have 

power to bring the rays proceeding from the pencil to 

the same focus with those from the object; this difficulty 
is obviated by the use of a convex lens at Z, or a con¬ 
cave one at L '; the effect of the former being to reduce 

the divergence of the rays from the pencil to the same 
degree with that of those from the object, and of the 
latter, to increase the divergence of the rays from the 
object, and render it the same with that of the rays from 
The instrument the pencil. The camera lucida is constructed of various 
forms, having reference to the facility of using it, the 
optical principle being the same in all. 


has various 
forma 











ELEMENTS OF OPTICS. 


263 


CAMERA OBSCURA. 

§ 88. This instrument is also used to copy from Camera ob3Cura; 
nature, and like the camera lucida, has various 
forms, one of the best of which is represented* in the 
figure. AJBCis & prismatic 
lens , which is nothing more 
than a triangular prism with 
one or both of its refracting 
faces ground to spherical sur¬ 
faces ; it is set in a small box 
resting on a cylindrical tube 
tv, that moves freely in a 
similar tube in the top of a 
dark chamber, formed by up¬ 
rights or legs, about which 
is suspended a cotton cloth 
rendered impervious to light 
by some opaque size. On one 
face of the box m n, contain¬ 
ing the prismatic lens, is an 
opening to admit the light 
from any object in front of 
the instrument, and on one side the cloth has been 
omitted in the figure to show a table X Y, supported 
by the uprights, on which the paper is placed to re¬ 
ceive the picture. Now, the rays from any point in an 
object /S, will enter the face A 0 of the prismatic lens, 
be totally reflected by A B, and brought by G B, to a 
focus on the paper, from which, owing to the minute 
irregularities of its surface, they will be reflected in all 
directions; and thus a picture of the external object S 
will be painted at S', which may easily be traced by a 
person situated within the folds of the cloth forming the 

... ° . Its action in 

dark chamber. The effect of the prismatic lens being form i ngimage8 
the same as that of a convex lens, except that the former explained; 


Fig. 62. 



Used to copy 
from nature ; 


Its essential 
parts; 












264 


NATURAL PHILOSOPHY. 


Construction of 
tbe image; 


Means of 
adjusting; 


Method of using. 


Magic lantern, 


Essential parts; 


Graphic 
representation; 


changes the direction of the 
axis of a pencil deviated by it, 
it is obvious that the surface of 
the paper should be spherical. 
The image of the object is 
brought accurately to the table 
X Y, by means of the tube t v, 
which admits of a vertical mo¬ 
tion in the top of the chamber; 
this tube also admits of a 
horizontal motion, the purpose 
of which is, to take in differ¬ 
ent objects in succession with¬ 
out changing the position of 
the body of the instrument. 


Fig. 62. 



THE MAGIC LANTERN. 

§ 89. This consists of a small close chamber, from 
one side of which proceeds a tube containing usually 
two convex lenses A and B , with an intermediate open¬ 
ing for a glass slide (7, which may be moved freely in 
a direction at right angles to the common axis of the 
lenses. Within the chamber is an Argand lamp Z>, 


Fig. 63. 



behind which is a concave reflector E. The rays pro- 





























ELEMENTS OF OPTICS. 


265 


ceeding from any point in a figure, painted with some °i> tical 
transparent pigment upon the glass slide and strongly 0,6 

illuminated by the lens A, upon which the direct light ex P lained ; 
from the lamp, as well as that from the reflector E, 
is concentrated, will be brought to a focus by the lens 
on a screen MN, placed at a distance in front of 
the instrument; here the light being reflected will pro¬ 
ceed as from a new radiant, and a magnified image of 
the figure will thus appear upon the screen. Should 
the screen be partially transparent, a portion of the light 
will be transmitted, and the image will be visible to an 
observer behind it. 

The linear dimensions of the object or figure, will Relation between 

. . . . the dimensions of 

be to those oi the image, as their respective distances the object and 
from the lens B ; if, therefore, the lens B be mounted image ; 
in a tube which admits of a free motion in that con¬ 
taining the lens A, its distance from the figure may be 
varied at pleasure, and the image on the screen made 
larger or smaller, the instrument, at the same time, be¬ 
ing so moved as to keep the screen in the conjugate 
corresponding to the focus occupied by the glass slide. 

The instrument with an arrangement by which this can 
be accomplished, is called the phantasmagoria. In or- phantasmagoria, 
der, however, that the deception may be complete, there 
must be some device to regulate the light, so that the 
illumination of the image may be increased with its 
increase of size, not diminished, as it would be without 
such contrivance. 


SOLAR MICROSCOPE. 


§ 90. This is the same as the magic lantern, except solar 
that the light of the sun is used instead of that from microscope » 



266 


NATURAL PHILOSOPHY. 


Solar 

microscope; 


a lamp. D JE, is a long reflector on the outside of a 
window shutter, in which there is a hole occupied by 
the tube containing the lenses. 


Fig. 64. 

t 


Essential parts 
and manner of 
using. 



The object to be exhibited is placed near the focus 
of the illuminating lens A, so as to be perfectly en¬ 
lightened and not burnt, which would be the case were 
it at the focus. 


CHROMATICS. 


Chromatics; 


Color in light 
corresponds to 
pitch and 
harmony in 
sound; 


Explanatory 

remark^; 


§ 91. Chromatics is a name given to that branch of 
optics which treats of Color. Color is to light what 
pitch and harmony are to sound. We have seen, in 
Acoustics, that by the principle of the coexistence and 
superposition of small motions, any number of sonorous 
waves may exist at the same time and place, and pro¬ 
duce, through the organs of hearing, an impression dif¬ 
ferent from that produced by either of the waves when 
acting singly. The united tones proceeding from the 
various voices of a full choir of music, for example, im¬ 
press the ear very differently from the insulated note 
of the acute treble, the medium tenor, or the full, 
deep-toned bass; and as each voice is partially or 
wholly suppressed in succession from a full strain of 
concordant sounds, while others are reinforced, the mind 














ELEMENTS OF OPTICS. 


267 


marks the change, and attributes to it a distinct and Analogy between 
specific character. the action «f 

So it is with the luminous waves which act upon the luminous waves; 
organs of sight. These come to us from the sun, and 
other self-luminous bodies, of every variety of length ca¬ 
pable of affecting the eye; they coexist and are super¬ 
posed upon the retina, and by their united influence 
give us the impression of white light; and when one white light 
after another of these waves is enfeebled, while others produced; 
are strengthened, each new combination gives us a dif¬ 
ferent impression, and each impression we call a color. 

The longest waves capable of affecting the eye corres¬ 
pond to red , and the shortest to motet or lavender grey. 

But how are individual waves either suppressed or 
separated from the group which produce the sensation Principles which 
of white light? The answer is, by the principles of in - producecolom- 
terference and of unegual refrangibilii/y. 


COLOR BY INTERFERENCE. 

Colors of Gratings . 

§ 92. Kecalling the expla¬ 
nation of §7, let MN be a 
wave front proceeding from 
a source 0. Assume any 
point O r , in front of the 
wave, and draw the straight 
line O r 0. Take the dis¬ 
tance A B equal to half the 
length X r , of. the longest, 
and A B' equal to half the 
length of the shortest 
wave capable of affecting 
the organs of sight; and make B C = CD = AB = construction of 
With O r as a centre, and the radii O r B\ O r B, O r C , fl?ure; 


Fig. 6!I 



Colors of 
gratings; 





268 


NATURAL PHILOSOPHY. 


colors of O r D^ &c., describe arcs 

gratings; cutting the wave front in 
5', 5, c, d , &c.; then will the 
portions A 5, 5 c, c^, &c., 
in the immediate vicinity 
of A, partially, and those 
remote from the same point, 

Construction of wholly, interfere, §7, and 
figure; neutralize each other’s ef¬ 

fects at O r ; for, at this point 
the secondary waves from 
the successive points of the 
portion A 5, beginning at A , will be opposed to those from 
^Taonsof^tin th e corresponding points in the portion i c, beginning at 
wave opposed to being in opposite phases; and it is plain that if the 
the odd portions. portions be numbered in order from A, that 

those distinguished by the even will be opposed to those 
designated by the odd numbers, the odd portions tending 
to displace the molecule at O r in one direction, and the 
even ones in the opposite direction, 
consequence of Now, conceive the even portions 5 c, de,& c., to be 

stopping the stopped by the interposition of some opaque screen; the 

even portions; r x 7 

odd portions no longer being neutralized, will have full 
effect upon <9 r , which will become greatly more lumi¬ 
nous by the conspiring action of the longest waves— 
Effect of longest that is, by the waves whose length is But the shortest 
increased^ o secon( ^ ai T way es proceeding to O r from the portion 5 5', 
on one end of A 5, will interfere and neutralize those 
from an equal portion A «, at the other end, so that the 
effect of the longest waves at <9 r , will be increased in 
a much greater proportion than that of the shortest. 

From the construction O r c — O r A — Take a point 
O v , such that O v c — O v A — then will' the point <9 r , 
for the same reasons, receive the greatest possible dis¬ 
turbance from the action of the shortest waves which are 
waves rnost° rteSt h ere the same phase, while the effect of the longest 
increased at o v ; waves, at this point, will be less than at *6> r , being no 
longer in the same phase. The effect of the waves whose 


Fig. 65. 







ELEMENTS OF OPTICS. 


269 


lengths are intermediate between X r and will have 
their preponderance upon molecules between these two 
points, and the space O r O v , should exhibit correspond¬ 
ing effects. And this is found by experiment to be the 
case. For when a grating is formed by fine parallel Effect30ffurr0W3 
[wires, or by a series of fine furrows cut in the face of mp es 1183 
a piece of well polished glass, and held in front of any 
luminous source, there will be formed upon a screen 
placed behind, a series of richly colored fringes, sepa¬ 
rated by dark intervals, and arranged along a line per¬ 
pendicular to the furrows. The furrows intercept the 
light, while the intermediate spaces between permit it 
to pass; the former correspond to the even and the lat¬ 
ter to the edd portions of the luminous wave referred to 
above, and form, as it were, a series of parallel linear 


Fig. 66. 



radiants. The molecules 0 r , <9 2r , &c., where the longest 
waves prevail, exhibit red, and those at O v , 0 2v , &c., Explanation or 
where the shortest preponderate, violet or lavender grey, the color9 ‘ 
the molecules between exhibiting orange, yellow, green, 
blue and indigo, in the order named, beginning at the 
red. The line X Y\ being drawn from the luminous 
source to the middle of an opaque portion of the grating, 
the first fringe on either side of this line is formed by 
secondary waves whose radii differ by X, the second by 
2 X, the third by 3 X, and so on; that is, every fringe is 










270 


NATURAL PHILOSOPHY. 


collateral fringes, formed by the conspiring of secondary waves whose radii 
differ by some even multiple of while the dark spaces 
between are produced by the opposition of waves of 
which the radii differ by some odd multiple of 


Effects of light 
transmitted 
through 
furrowed glass; 


Effects of light 
reflected from 
the same; 


Fraunhofer and 

Barton’s 

experiments. 


§ 93. If the furrowed 
glass be interposed between 
the eye at 0 , and any lu¬ 
minous source A, say a 
small hole in a window 
shutter, the latter will ap¬ 
pear flanked on either side 
by similar fringes with in¬ 
termediate dark spaces 
between also arranged on 
a right line perpendicular 
to the direction of the fur¬ 
rows, the red appearing on 
the outside, the violet on 
the inside, with the other 
colours in the order just 
named between. And to 
an eye at 0\ so placed as 
to receive the light reflect¬ 
ed from the ridges of plane 
glass between the furrows, tl 
appear covered by the most 
change with every change o 


Fig. 67. 



0 


ie whole furrowed space will 
beautiful irised hues which 
f position of the eye. 



By means of a fine diamond point, Fraunhofer suc¬ 
ceeded in forming a ruled surface of glass in which the 
striae were actually invisible under the most powerful 
microscope, the interval of the furrows being only 
of an inch. In some furrowed surfaces produced by Mr. 
Barton, the lines are so close that 10000 of them would 
occupy only the space of an inch in breadth. The light 
reflected from surfaces so minutely divided, exhibits the 
purest colors of which we have any knowledge. Simi¬ 
lar appearances are exhibited when light is reflected 









ELEMENTS OF OPTICS. 


271 


from metallic surfaces which have been polished by a 
coarse powder, and from surfaces of glass over which 
the finger is passed after being moistened by the breath. 

The beautiful colors of mother of pearl are natural Effects of the 
instances of the same phenomena. This substance is^ i “^“ mother 
composed of a vast number of thin layers, which are 
gradually and successively deposited within the shell of 
the oyster, each layer taking the form of the preceding. 

When it is wrought, therefore, the natural joints are cut 
through in a great number of sinuous lines, and the result¬ 
ing surface, however highly polished, is covered by an 
immense number of undulating ridges formed by the out¬ 
cropping edges of the layers. These striae may be ob¬ 
served by the aid of a powerful microscope, although they 
are so close that 5000 of them occupy but a single inch. 

That they are the cause of the brilliant colors displayed Experiment »r 
by this substance has been placed beyond doubt by Sir lister 
David Brewster, who received from an impression of the 
surface of pearl on soft wax the same display of colors 
as from the pearl itself. 

§94. Knowing the space A <7, occupied by a single to find the 
furrow and one of its adjacent transparent intervals, itluminouswavw 
wall be easy to find the lengths of the waves which cor¬ 
respond to the different colors. For this purpose we 
remark, 

1. That the first 
colored fringe F x , 
seen through the 
glass on either side 
of the luminous 
source L is formed 
by secondary waves 
wh ose rad i i d iffer 
by X, the second F % 
by 2 X, the third by 
3 X and so on, and 
are called respec- 


Remark 1st. 


Fig. 69. 









272 


NATURAL PHILOSOPHY. 


Fringes of 
different orders. 

Eemark 2d. 


Distance of the 
fringe of any 
order. 


Explanation. 


Wave length. 


tively fringes of the 
first, second , third , 

&c., order. 

2. The angular 
distance of a fringe 
of any order from 
the luminous source 
Z,is sensibly equal 
to the angular dis¬ 
tance of the fringe 
of the first order, 
multiplied by the 
number which marks the order of that fringe. For ex¬ 
ample, the angular distance LEE 3 of the fringe of 
the 3d order, is equal to the angular distance LE F y 
multiplied by 3. 

The actual distance of the projection of the fringe of 
any given order from the luminous source will be pro¬ 
portional to the distance of the furrowed glass from the 
•same source. I F x , for instance, is obviously greater 
than E FI, and is proportional to E Z. 

Now, with centre E and radius E A, describe the arc 
A K. The distance GK will be equal to X, or the 
length of the wave corresponding to the color seen along 
the line E C. 

Denote the distance EA by r; the sum of the furrowed 
and transparent parts, which is equal to A C, by 5 ; the 
angle AEC by <5; and the length of the wave, equal to 
EC, by X; then will 

$ 2 = X (2 r + X); 

and because X may be neglected in comparison with 2 r, 
the above may be written, 

X = t \ s . tan 5 ; 

and because, for the small angle <5, the arc may be taken 
for its tangent, 


Fig. 69. 



Bam®. 


X = L s . 8 


(85) 













ELEMENTS OF OPTICS. 


273 


From which it appears that <5 will be greater in pro¬ 
portion as X is greater; and that the colors of each 
fringe which correspond to the longest waves will, there¬ 
fore, be found at the greatest distance from the lumin¬ 
ous source. 

To find X, we have only to find s and S . The for¬ 
mer is known from the cutting of the furrows, and M. 
Babinet has given a very simple method for determin¬ 
ing the latter. It is as follows. 


Fig. 70. 



§ 95. Let A and B be two 
candles, or still better, two nar¬ 
row openings in a plate of metal 
held before a window, the one, B , 
a little more elevated than the 
other; Bp, a perpendicular to 
A B at its middle point P, and 
B B\ the furrowed glass placed 
somewhere upon P]), and held so 
that the furrows shall be per¬ 
pendicular to A B. On placing 
the eye at P, and looking through 
the glass, we shall see the two 
candles or openings by direct 
view, and each of them flanked 

on either side by a series of fringes, those about B be¬ 
ing higher than those about A. By moving the glass 
B B\ to or from the candles, the red, or any other co¬ 
lor of the fringes on the left of P, may be brought 
accurately over the same color of the fringes to the 
right of A. These coincidences, however numerous, may 
be established because of the law of the angular and 
true distances of the fringes from their respective lumi¬ 
nous sources referred to in the preceding article, 
provided the angle B E A, does not exceed 5 or 6 de¬ 
grees. 

Denote by n , the number of coincidences of any par¬ 
ticular color between A and P; there will be n + 1 
18 


Colors 

corresponding to 
longest waves 
furthest from 
luminous source. 


To find \ 
experimentally, 


Explanation of 
apparatus; 


Coincidences of 
same color 
produced; 


Notation; 





274 


NATURAL PHILOSOPHY. 


\ 

Graphic 
representation ; 


Proportion from 
figure; 


Same in form of 
an equation; 


Equation for 
email angle; 


Notation ; 


Substitution; 


First form of 
value for X 5 


fringes from A to in estimat¬ 
ing from A, for one of the fringes 
belonging to A will cover or fall 
upon the candle B. The angle 
ABB will, therefore, be equal 
to (n 4-1). 8 ; and its half PEA, 

to n - J( ~ ^ . 6. But in the right 
2 

angled triangle PEA , we have 

EP:P A :: 1: tan w + 1 . 6. 

9, 


whence 


Fig. 70. 



tan 


n +1 

' 2 


. <$ 


PA 
~ PE * 


As long as PEA does not exceed two or three degrees, 
its tangent may be taken equal to its arc, and we may 
write 


n + 1 ^ _ P A 

‘ ~ PE ' 


whence, denoting AB by Z>, and PE by d, 

a=_J_. 

{n 4-1). d 

and this value in Equation (85), gives 


2 X — 


s.b 

(n + 1) . d 


. . . ( 86 ) 













ELEMENTS OF OPTICS. 


275 


and denoting by <?, the number of furrows in an inch, we Notation ; 
shall have 



and this in Equation (86), finally gives 


Substitution; 


, .( 37 s Final Taluo lb, 

c. (n -f 1). d X; 

whence this rule, viz: Augment the number of coinci¬ 
dences between the candles by unity • multiply this sum 
by the number of furrows in an inch , and this product 
by the distance , in inches , of the glass from the plane 
the candles , and divide the distance , m inches , between 
the candles by this product; the quotient will be double 
the length , m inches , of the wave , corresponding to the color 
with reference to which the coincidences are made. 

By the application of this rule in the manner indicated, 
we find, when the experiments are made in the atmos¬ 
phere, the results in the following 

• 


TABLE. 


Colors. 

Length of 

X 

in parts of an inch. 

Number of 

X 

in one inch. 

Extreme red, - - - - 

Orange,. 

Yellow,. 

Green,. 

Blue, -. 

Indigo,. 

Extreme violet, - - - 

0,0000266 

0,0000240 

0,0000227 

0,0000211 

0,0000196 

0,0000185 

0,0000167 

37640 

41610 

44000 

47460 

51110 

54070 

59750 













276 


NATURAL PHILOSOPHY. 


To find tho 
distance of any 
particular fringe 
from the central 
line; 


Illustration; 


Equations from 
the figure; 


Difference 
between these 
equations; 


Equal to n \; 


Value for 
distance of any 
fringe from 
central line. 


§ 96. Let us now determine the distances of the fringes 
from the central line X Y. They 
depend upon the sum A C = s, of 
the width of one opaque and one 
adjacent transparent interval, and 
upon the distance X Y\ of the 
screen from the grating. 

The place 0 , of the fringe of any 
order, say the nth, is determined 
by the condition that the difference 
of its distances A 0 and G 0 , from 
A and (7, is an integer multiple of 
the length X, of the wave of the particular color con¬ 
sidered. Now, drawing the lines A A' and C G\ paral¬ 
lel to X Y j and denoting the distance X Y by d, and 
Y 0 , by a?, we find 


Fig. 71. 
A X C 


l\ 

1 

\ 

1 

1 

1 

1 

1 

i 

ALA J 

f C’ O AT 


AO= ^ +(a;+ . d + 

the distance d , being very great in comparison with x 
and s. 

Hence, # 

4 0 — o 0 = i s ) 2 ~~ 0** ~~ £ S Y s • x m 

2 d d 

But this difference is equal to n X; that is, 


whence 


x — 


n A .d 


s 


( 88 ) 














ELEMENTS OF OPTICS. 


m 


From which it appears that that the fringes will be crowd- conditions than 
ed together more and more in proportion as d decreases wil1 cau8e the 

0 lx fringes to crowd 


\ , lringes to cr< 

relatively to s, or s increases relatively to d —a result towards the 


confirmed by experience, for when the screen is either centre - 
made to approach the grating, or the furrows are in¬ 
creased in size, the fringes will be observed to contract 
and crowd in upon the centre Y, till the^ become so 
narrow as not to be perceptible. 

§ 97. Again, let X r and X r , denote the lengths of the 
waves which give red and violet colors respectively; 
then will Equation (88) give 


n. d. \ 


Distances of the 
red and violet 
colors of the rcth 
fringe from the 
centre. 


S 


x, = 


S 


In which, because X r is greater than X c , a? r , which de¬ 
notes the distance from Y to the red color of the nth. 
fringe, will be greater than a? c , which represents the dis¬ 
tance of the corresponding violet color from the same 
point. 

Subtracting the second-from the first, we. get The colors 


separate from 
each' other; 



(89) 


From which we see that the different colors will sep¬ 
arate more and more as the fringe to which they belong And t h 0 dark 
recedes from the centre Y. The black intervals will, intorvals ftaal{ J r 
therefore, be encroached upon, and at no great distance 
from Y will disappear. 

To find the order of the last insulated fringe, denote by 
a? n+1 and x n the distances of the (w-fl)th and nth fringes oMhe^i 0 ^ 
from Y\ and by \ the length of the wave for red; then insulated fringe; 
will Equation (88) give 





278 


NATURAL PHILOSOPHY. 


Notation and 
equations ; 


Interval between 
the reds of two 
consecutive 
fringes; 


Order of the last 
insulated fringe. 


Experimental 

illustration. 


Experiment 
performed in 
vacuum ; 


«VH = 


(71 + 1 ) . X r . 




n . \ . d 


whence, taking the difference, we obtain for the inter¬ 
val between the reds of two consecutive fringes, 

\ . d 

x n+1 - x n = - 

s 

and placing the second member of this Equation and 
that of Equation (89) equal, we find 


n . d 


(X _ X ) = 

' r v> 


d 


whence 



That is to say, the order of the last insulated fringe is 
denoted by the number of units in the quotient arising 
from dividing the length of the red wave by the dif¬ 
ference of the lengths of the red and violet waves. 

This result is beautifully illustrated by interposing 
between the screen and grating some medium which 
will arrest all the waves but those which correspond to 
a particular color. When this is done, the fringes will 
be greatly multiplied in number beyond that of the Titli 
order determined by Equation (90). 

§ 98. Thus far the waves have been supposed to pro¬ 
ceed, after passing the grating, in the atmosphere. But 
when the experiment is performed in vacuum, with the 
same grating and same position of the screen, the fringes 
are found to dilate and separate from each other; when per 
formed in a medium of greater density than the air, as 









ELEMENTS OF OPTICS. 


279 


in water or glass, tlie fringes are reduced in width and Experiments 
crowded towards the centre; and what is remarkable, adln^mldium 
and important to observe, this latter effect is found, by thanair ; 
careful experiments, to be exactly proportional to the in¬ 
dex of refraction of the medium as referred to that of 
atmospheric air . 

How, referring to Equation (88), it is easy to see that 
this change in the position and width of the fringes Causeofthe 
can only arise from a change in X, which denotes the position and 
length of the waves, since s and d are, by the conditions width of th « 
of the experiment, constant; and from the relations 0 f fnn ° es ’ 
x and X, in that Equation, it follows that the length of 
luminous waves of the same color are shorter in propor¬ 
tion as the indexes of refraction of the media in which 


they exist are greater. But, Equation (2), these indexes Lengths of w« 
vary inversely as the velocities of wave propagation, and in different 
hence the lengths of the waves are directly proportional 
to the velocities with which they are transmitted through 
different media . The cause by which the lengths of the 
waves are thus altered in the direct proportion to their Princ5ple ofw 

, x , acceleration a 

velocities, is called the principle of wave acceleration retardation. 
and retardation. 




§ 99. Returning to the experiment in air; if a very ^ ^ 
thin plate of glass be interposed in front of one of the interposing a 
grate openings, and parallel to the plane of the grating, plate of gla8S 

r & > r o’ under different 

the whole system of fringes will be shifted towards conditions, 
the side of the interposed glass. If an exactly 
similar plate be placed in front of the other opening, 
and parallel to the first plate, the fringes will be re¬ 
stored to their original position. If one of the plates 
be slightly inclined, so as to cause the waves passing 
through it to traverse a greater thickness, the fringes 
will all move towards that side, and by gradually in¬ 
creasing the inclination, they will pass entirely out of 
sight. 

Taking plates of any other medium, possessing a 
greater refractive index than glass, and of the same 



280 


NATURAL PHILOSOPHY. 


Effect of 
Interposing a 
plate of any 
medium. 

Effect on the 
lengths of the 
rays; 


This last effept 
investigated 


Illustration; 


Explanation; 


Equation from 
figure; 


thickness as before, it is found that the effects just no¬ 
ticed will be increased, and in the direct ratio of the 
refractive indexes of the media. 

In the shifting of the fringes, it is evident that the 
lengths of the rajs which correspond to the central one 
are made unequal, and that the differences as to lengths 
existing among the rays which appertain to the other 
fringes, are not the same as before the interposition of 
the medium. We will now investigate this change. 

For this purpose, let the waves from both openings 
pass through a prism of any medium, as glass, hav¬ 
ing a very small refracting angle, 
i, the first face being held pa¬ 
rallel to the plane of the grating. 

The thickness of the prism tra¬ 
versed by two interfering waves 
will be different; call this diffe¬ 
rence, which is r n in the figure, 
d. Draw n n', parallel to KL\ 
with 0 , as a centre and 0 r as a 
radius, describe the arc rr\ It 
is obvious that the number of 
waves in the length An + Or will 

be equal to the number in the length Cn'+Or', since 
the circumstances are the same in both routes; the only 
difference, if there be any, must lie in the paths n r 
and n’ r'. Since the angle made by the rays A 0 and 
CO, is very small, these rays will enter the first sur¬ 
face under very small angles of incidence, and both be¬ 
ing refracted towards the perpendicular, their direction 
through the prism will be nearly normal to that surface; 
hence, denoting by b, the distance r n', we have 

\ ■. . . 

a = o . Bin i : 



but under the above supposition, the angle of incidence 
at the second surface will be equal to i; and denoting 







ELEMENTS OF OPTICS. 

the corresponding angle of refraction by 9 , we also get 
sin 9 = m sin i ; - 

in which m is the relative index for glass. 

Denoting the distance n' r' by d\ we have, because 
r r f is very small, 


d' = b. sin 9 = b. m .sin i = m d = -y T • d 


in which V denotes the velocity of the waves within 
the air, and V' that in the glass, and from which we 
find 



did':: V': V; 


Proportion; 


Conclusion 


that is, the distances r n and r' n\ are directly propor¬ 
tional to the velocities with which they are traversed by 
the waves; they must, therefore, be passed over in the 
same time, and the velocity in air will exceed that in glass ,— 
a fact fatal to the Newtonian or emission theory of light, 
which requires the converse to be true, and which for a 
long time contested the claims of its rival hypothesis of 
wavesj first advanced by the celebrated Hcyghens. There 
will be the same number of waves in nr as in nV, and 
while the lengths of the routes from A and C to 0 will 
differ when expressed in the same unit, yet these routes, 
estimated by the number of waves in each, will be equal. 

Before leaving this subject, it may be remarked, that 0onfirmi * ion o1 
the lengths of waves answering to different colors have the tabular 
been computed by means of Equation ( 88 ), after care¬ 
fully measuring the distances x and d , and were found 
to agree in all respects with the results given in the 
table of § 95. 


ref-ults of § 95. 



282 


NATURAL PHILOSOPHY. 


COLORED FRINGES OF SHADOWS AND APERTURES. 


three fringes 


“iciis § 100. When an object is placed in a pencil, such as 
of light usually ma y k e formed by admitting light through a very small 
bordered by a p e rture into a dark chamber, or by a convex lens, and 
the shadow of the object is received upon a screen, it 
is found to be bordered externally by fringes, usually 
three in number, at decreasing distances from each other, 
each fringe being made up of different colors. These 
fringes are parallel to the outlines of the shadow, ex¬ 
cept when the latter terminate in a salient angle, in 
which case they curve around it; or when the outlines 
form a re-entering angle, and then the fringes cross and 
run up to the shadow on each side. 

In white light, the colors of the fringes, reckoning from 
the shadow, are, in the first, black, violet, deep blue, 
light blue, green, yellow, and red; in the second, blue, 
yellow, and red; and in the third, pale blue, pale yel¬ 
low, and pale red. 

In homogeneous light, the fringes increase in number 
and are alternately dark and bright. In passing from 
one color to another, they vary in width, being broadest 
in red and narrowest in violet; and it is from the par¬ 
tial superposition of these and the remaining colors, that 
the different colors arise when the experiment is made 


Positions of 
these fringes. 


Colors <|f the 
fringes in white 
light. 


Peculiarities of 
the fringes in 
homogeneous 
light \ 


Fringes 


with white light. 


These fringes are entirely independent of the nature 
the nature of the an d figure of the body whose shadow they surround, 
body. being the same when formed by a mass of platina or a 

bubble of air—by the back or edge of a razor. 
Measurements of The shadow being received upon a convex lens, be- 
thofnn e esshow, which is placed a micrometer, the linear elements 
of the fringes may be measured to any desired degree 
of accuracy. These measurements show : 

First ; 1st. That the distances between the fringes and the 

O 

shadow diminish as the lens approaches the body, and 






ELEMENTS OF OPTICS. 


283 


finally vanish, so that the fringes have their origin 
close to the edge of the body. 

2 d. That the locus of each fringe is an hyperboloid second; 
of revolution, terminating near the edge of the body. 

3d. That, the distance of the lens from the body re- Tbird . 
maining the same, the fringes will be more dilated, as 
the body approaches the luminous point. 

It is also found, that when the luminous point is in- Effect 
creased so as to become an appreciable circle, the fringes increasing the 
formed by the light proceeding from each of its points luminous poi f* 
overlap and confuse one another, obliterating the colors 
and forming a penumbra, which consists of a ring whose 
brightness varies from the edge of the shadow, where 
it is least, to its exterior boundary, where it is greatest. 


§101. If the size of the body be much reduced in Effect of 
one direction, parallel to the screen or plane of the lens, r f du ® in ^ tbo j 
the shadow will be found to consist of bright and dark parallel to the 
fringes parallel to the length of the body, a bright fringe screen ; 
occupying the centre. If the body be small and of a 
circular form, having its plane parallel to that of the 
screen, the shadow will be made up of a series of con¬ 
centric bright and dark circles, having a bright spot in 
their centre. 

As the body diminishes in size, the stripes diminish Appearances L 
in number and increase in width, till all disappear but the bodjr 

diminishes. 

the central illumination. The reverse effect will arise 
either on increasing the size of the body, or diminish¬ 
ing its distance from the screen. 


§ 102. If a portion of the pencil be transmitted through I>eiiciI admitted 
a small and well defined circular aperture and received 
upon a screen, concentric rings will also be produced; 
and if the transmitted portion be viewed through a con¬ 
vex lens, the hole will appear as a bright spot, encircled 
by rings of the most vivid colors, which undergo a .great 
variety of changes, both as regards tint and linear dimen¬ 
sions, in varying the distance of the lens from the aper- 



284 


NATURAL PHILOSOPHY. 


ture, and that of the aperture from the radiant or lumi¬ 
nous point. 

Light When the light is transmitted through two very small 

through two apertures, close together, rings corresponding to each 
circular apertures w [\\ p e formed as before, and in addition there will be 

close together. 4 9 

found a number of straight parallel fringes between the 
centres of the circles, and at right angles to the line 
joining them; two other sets of parallel fringes will also 
be seen in the form of St. Andrew’s cross proceeding 
from the space between the centres; and by multiplying 
the number of the apertures and varying their relative 
dimensions, a set of phenomena arise of exceeding bril¬ 
liancy and beauty. 

Cause of the § 103. The colored fringes of shadows and small aper- 
coiored fringes of tures, as well as all appearances referred to under this 
apcrtiirea. an<i head, are caused by interference '*the interference taking 
place between the secondary waves from the edges of the 
body or aperture and those from that portion of the 
primitive wave which is not intercepted. 

Names originally These phenomena were at one time called the inflection 
apphedjto them. Qr diffraction light ? and were supposed to arise from 

some peculiar action exerted by the edges of bodies on 
the rays as they passed near them. 

Effector If the refractive index of the medium in which the 

increasing the experiments are performed be increased, the phenomena 
of the medium, indicate a diminution in the lengths of the waves in the 
same ratio. 

I 

COLORS OF THIN PLATES. 

All media § 104. Transparent, and indeed all media, when ro 

exhibit colors duced to very thin films, are found to exhibit colors 

thin films which vary with the thickness of the film. These are 

called, the colors of thin plates , and the easiest way to 
exhibit them is by means of a soap bubble blown from 
the end of a quill or the bowl of a common smoking 

*Seo Appendix No. 1. 



I 


ELEMENTS OF OPTICS. 285 


pipe. As the bubble increases in diameter, and the Familiar 
fluid envelope is reduced in thickness at the top by Ma* 6 
gradual subsidence toward the bottom, many colored plates; 
and concentric rings will be seen around the point of 
least thickness. At this point, the color will be found 
(do change, first appearing white, then passing through 
blue to perfect blackness, the rings the while dilating till 
the bubble is destroyed. 

The same is true of any other medium, whether gase¬ 
ous, liquid, or solid. 

These different colors being exhibited upon the same when the plate 
plate of variable thickness, no single color can be iden- “ 

tified with its chemical composition. When of uniform 
thickness, a single color only will be seen, and this will 
change as the thickness of the plate changes. 

A thin plate is very con¬ 
veniently formed of air; and 
for this purpose, let A B , be 
a plano-convex, and CD a 
plano-concave lens, placed 

one upon the other, as represented in the figure. When 
this arrangement is viewed on either of the plane faces 
by reflected light, colors will be seen in the form of con- Appearances by 
centric circles about the point of contact, which, should reflectedlight; 
the pressure be sufficient, will be totally black. If view¬ 
ed by transmitted light, rings whose colors when 
united with those of the first, form white light, and which By transmitted 
colors are, therefore, said to be complementary , will ap- light; 
pear about the central spot, which will now be per¬ 
fectly white. With waves of a single length, as yellow, 
these rings are alternately bright and dark, begin¬ 
ning with the central spot; and by reflected light, 
dark and bright. They are broadest and have the great¬ 
est diameter in the red, and narrowest with least diam¬ 
eter in the violet; the breadths and diameters in the 

\ t ' Effects due to 

other colors being intermediate and varying in magni- waves of a single 
tude in the order of the spectrum from red to violet. It leugth; 
is by the superposition of these rings, or the waves 









286 


NATURAL PHILOSOPHY. 


Newton’s scale/ which produce them, that the different colors appear in 
common light. 

These colors, which are of different orders as regards 
tint, constitute what is called Newton’s scale; and by 
reflected light, occur as follows, beginning with the cen¬ 
tral spot. 



First order; 
Second; 
Third; 

Fourth, Asp,; 

V 


Waves which 
interfere to 
produce the 
colors by 
reflexion; 


Illustration and 
explanation; 


1st order. Black, very faint blue, brilliant white, yel¬ 
low, orange and red. 

2d order. Dark violet, blue, yellow-green, bright yel¬ 
low, crimson and red. 

3d order. Purple, blue, rich green, fine yellow, pink 
and crimson. 

Hh order. Dull blue-green, pale yellow-pink, and red. 

5th order. Pale blue-green, white and pink. 

3th order. Pale blue-green, pale pink. 

7th order. The same as 6th, very faint. The other or¬ 
ders being too faint to be distinguishable. 

These colors arise from the in¬ 
terference of waves reflected from 

. 

the first, with those reflected from 
the second surface of the air 
plate. 

Suppose a small beam incident 
perpendicularly or nearly so, on 
the first surface MN of the plate, 
where the thickness is t. A part 
A O will be reflected back, the 
rest A B , being transmitted, w T ill 
traverse the thickness t. At the second surface, again 
a part B G , is reflected, and the reflected portion return¬ 
ing through the thickness £, will emerge at the first sur¬ 
face in the direction C 0 , and be superposed on that first 
reflected at this surface, and these will either conspire 
and reinforce each other or will interfere and partially 
or wfiolly neutralize each other, according to any of the 
conditions explained in § 7, depending upon the differ- 







ELEMENTS OF OPTICS. 


287 


enee of route 2 t. Whenever 2 t is equal to any even if 2 1 be an even 
multiple of \ X, for any color, this color will be increased, mnltlple of * X; 
and when equal to any odd multiple of \ X, it will be 
suppressed. ]STow, 2 t y will vary from a value sensibly 
nothing to one equal to many times X, for even the long¬ 
est waves, in passing outward from the point in which 
the spherical surfaces are tangent to each other, and if an odd ^ 
hence the colored fringes and the intermediate dark mult,ple ‘ \ 
rings. 

But the portion reflected at the second surface will, Transmitted 
in part, be again reflected at the first, and will traverse ”3 S*ctid 
the thickness t , a third time, and emerge below super- rin gs are darii 
posed upon the portion first transmitted at the second 
surface. The difference of route of these portions will 
also be 2 £, so that the effects should be the same on 
either side of the lenses. Experiment shows, however, 
that this is not the case, for wherever there is total 
darkness by reflexion, there is a maximum of bright¬ 
ness by transmission. Hence, there must be half a wave 
lenqth subtracted from the route at each internal re- This differcnce 

* .... accounted for: 

flexion / the cause of the loss being a change in density 
and elasticity at the surfaces of contact of the glass and 
air. This will give for the interfering rays, in case of 
reflected rings, a difference of route expressed by 


2 1 + —; 
2 


Difference of 
route for refleci 
rings; 


Fig. 75. 


and for the transmitted, 

2 t + X. 

To ascertain the value of t , at the 
different rings, call d , the diameter 
2 PH ’ of one of them, as determined 
by actual measurement; r and / the 
radii of the surfaces, v and v\ the cor¬ 
responding versed sines of the arcs whose sines PII and 
P' 27', are equal to the semi-diameter of the ring in question. 



Same for 
transmitted ring& 


To find r, at the 
different lings; 






268 


NATURAL PHILOSOPHY. 


ten, for very small arcs, we have 


Equation from 
the figure; 


ar 

9 = T7 i 


Another 
equation; 


,jd: 


2 r' 


Value of t- 


whence 


, d 2 (1 1\ 

= v ' - v = — I—-). 

8 \r r / 


same for first In this way Newton found the thickness at the brightest 
bnght nng ; p ar j; 0 f the first ring nearest the central black spot, to be 

0,00000561 of an inch. He also found the diameters of 

Law of variation . 

of diameters of the darkest rings to be as the square roots of the even 
dark and bright numbers 0,2,4,6, &c., and those of the brightest as 

rings; 


the square roots of the odd numbers 1,3,5,7,, &c. 
ffhe radii of the surfaces being great compared with the 
diameters of the rings, the value of t at the alternate 
points of greatest obscurity and illumination are as the 
natural numbers 


Law of variation 
of t, at the dark 
and bright rings; 

Eule. 


0,1 , 2 ,3,4 , &c., 


Above results 
compared with 
X for yellow; 


lienee, the value of just found, multiplied by these num¬ 
bers, will give the thickness at the different rings. 

On comparing the value for the thickness at the first 
bright ring, with the numbers in the table of article 
(95), it 'vfrill be found just equal to one-fourth of the 
interval denoted by A, for the yellow' ray, which is the 
most illuminating of the elements of white light. 

Taking this value for t , we shall have for the difference 



ELEMENTS OF OPTICS. 


289 


\ 


of route for the interfering rajs producing the dark Difference of 
rings bj reflexion, including the central black spot, 



rings; 


2 ’ 2 ’ 2 9 2 ’ *’ 


X 3 X 5 X 7 X Q 

-r-, &C. 


these being the even multiples of \ X, increased b j \ X 
for the retardation caused by one internal reflexion. 
The odd multiples, increased by \ \ give 


X, 2 X, 3 X, &c. 


Same for the 
bright reflected 
rings. 


The transmitted rings will be complementary to those 
seen by reflexion. 

The phenomena we have just considered are equally same 
produced, whatever may be the medium interposed P henomM,a 
between the glasses, the only difference being in the different media, 
contraction or expansion of the rings, depending upon 
the refractive index of the medium. It is found that as 
the refractive index of the medium increases, the diame¬ 
ter of the rings will decrease, which might have been 
inferred from article (99). 

§ 105. If any one of the rings at a particular color be 
conceived to be expanded in all directions in the plane of 
the ring arid to retain the same thickness, it is obvious 
that the plate thus produced would present the same 
color over its entire surface. If a second plate of the colors of natnmi 


bodies explained. 


same thickness and material be placed behind this one, it 


would act upon the waves transmitted through the first 
just as the latter did upon the incident waves, and the 
same would be true of any number of plates, so that a 
body made up of a series of such plates would present 
a uniform, distinct, and characteristic color. These con¬ 
siderations, in connection with those relating to the 
color of minute gratings or striae, furnish an explana¬ 
tion of the colors of natural bodies. 



290 


NATURAL PHILOSOPHY. 


Colors of 
inclined glass 
plates; 


Circumstances 
attending the 
deviation of 
light by such 
plates; 


Emergent waves 
will generally 
have travelled 
routes differing 
in length ; 


Two will emerge 
after having 
travelled 
different routes 
of equal length; 


Illustration; 



ORS OF INCLINED GLASS PLATES 


)6. If a luminous object be viewed through two 
plates of glass of precisely equal thickness, slightly in¬ 
clined to each other, it will be evident that besides the 
transmitted image, there will be a number of images 
formed by the successive reflexions between the glasses. 
The first or brightest of these is formed by the waves 
which have all undergone two reflexions and at different 
pairs of the four surfaces. On entering the first plate 
they undergo a partial reflexion at every surface they 
successively encounter, each of the reflected waves un¬ 
dergoing a similar series of partial reflections at each 
surface. Thus it will appear evident that the different 
portions into which the waves have been separated must 
ojo through a length of route differing by the length of 
the interval between the glasses and the thickness of 
the glasses, or the different multiples of those which 
they have respectively traversed. They will, therefore, 
in general , emerge after traversing routes which differ 
by considerable quantities. 

\Among these portions, however, there are two which, 
(if we abstract the very small difference in the in¬ 
terval between the glasses at the points where they re¬ 
spectively pass,) will have gone through different routes 
of precisely equal length . These two w r aves will be, 

1 st. One which passes di¬ 
rectly through the first 
plate A B , equal to t, and 
through the interval B C, 
equal to i, between the 
plates, is then reflected 
at C, in the first surface 
of the second plate, re¬ 
turns along CD, equal to 
i , and a thickness D E, 
equal to the first, or t\ 


Fig. 76. 







ELEMENTS OF OPTICS. 


291 


at the first surface it is reflected again and passes the 
whole system EF + FG + G H, equal to 2 t + i\ 0 r Explanati01i; 
upon the whole, it has travelled over 4 t + 3 i. 

2 d. Another portion proceeds directly through the 
whole AF + FC+Cd, equal to 2 14 * i, is reflected at 
d , in the last surface, retraces the distance de + <?/, 
equal to t 4- i y is reflected at the second surface of the first 
glass and pursuing the course f g ~\r g A, equal to i + £, 
emerges after having, on the whole, passed through 
4 t -f 3 or a route exactly equal in length to that of 
the former, neglecting, as before, the difference in i. 

It will be seen that out of all the possible combina- No other waves 
tions of different successive reflexions, these two are the ^” d ^ thls 
only ones which will give routes precisely equal; all the 
others will differ by quantities amounting to some mul¬ 
tiple of t or L If we now recur to the small difference 
in the interval % for the points at which the rays respec¬ 
tively pass, it is obvious that by slightly altering the in¬ 
clination of the plates we may diminish the difference of 
routes to any amount, and may consequently make them colored fringe* 
differ by half a wave length, or any multiple of the 
same; and we shall thus produce colored fringes sepa¬ 
rated by dark bands, parallel to the intersection of the 
planes of the glasses. 


COLORS OF THICK PLATES, 


§ 107. Another phenomenon, which 
depends upon the same principle, 
and called the colors of thick plates , 
will be readily understood from pre¬ 
ceding considerations. 

The effect is observed to take 
place under these circumstances, 
viz.: Light being transmitted through 
a small hole A , in a screen, and al¬ 
lowed to fall upon a spherical con- 


Fig. 77. 



Colors of thick 
plates; 


How they may 
be exhibited; 






292 


NATURAL PHILOSOPHY. 


Illustration and 
explanation; 


Facts with 
regard to these 
colors; 


How produced. 


Notation; 


Equation; 


Another; 


Difference of 
routes equal to 
some multiple of 
r X. 



C ' 

■*!■ 



cave glass Reflector M with con- Fig. 77. 

centric surfaces, the back being sil¬ 
vered, and its centre of curvature 
situated at the aperture, there will 
be formed upon the screen about 
the aperture a series of colored, rings , 
or in luminous waves of a single 
length, alternate bright and dark cir¬ 
cles. these become faint and disap- ____ 

pear if the distance of the screen 
be increased or diminished! beyond a small difference 
from its original position. They diminish in diameter 
as the glass is thicker. They arise from the interference 
of waves which emerge from different points of the first, 
after being reflected from the second surface. 

Denote by y, the radius A D , of one of the rings, either 
dark or bright; by t, the thickness CE y of the reflector; 
and by r, the radius A C. The equivalent interval to t , 
in air, will be m t , in which m denotes the relative index 
of refraction for air and glass. The question is to find 
the difference of the routes 


AC + CE+EC+CD, 
AC+CE+ED; 

EC+ CD - ED ; 

EC+ CD = yV 2 4 - v 2 = + r + 

* 2 r’ 

by neglecting the fourth and higher powers of y ; and 


and 

or to find, 

Now> 


A , i> = V(8m < + r)»+y*=a»K + r + 2^ r j^; 

whence, 

EC+CD -ED= y ~~ y 


2 r 


1 













ELEMENTS OF OPTICS. 


293 


But this difference of route must be equal to some mul¬ 
tiple of | X ; whence, 



V‘ 


y 4 


2 r + r) 


= In. A. 


Value for radius 
of the assumed 
ring; 


and neglecting 2 m t in comparison with r, in value of y, 
we find, 


2mt 


(91) 


Same reduced. 


This accords precisely with the most exact measure¬ 
ments of Sir Isaac Newton. 



COLOR FROM UNEQUAL R 

§ 108. It is demonstrated in the “Analytical Mechanics,” 
§ 316, that the velocity of wave propagation through an 
elastic medium, is given by the equation 


V 2 = II * 


sin 


. Ar 
X~ 



(92) 


Velocity of wave 
propagation. 


in which V denotes the velocity of wave propagation, H a 
function of the elasticity and density of the medium, r the 
distance between the adjacent molecules, a r the projection 
of this distance on the direction of the wave motion, and 
A the wave length. 

Now, when r is very small in comparison to X, the arc of 
which the sine enters the last factor above will be small; 
the ratio of the sine to its arc will be equal to unity, and 
the velocity will be simply equal to VII. In other words, velocity win be 
when the distances between the consecutive molecules of the8aniefor 

.. .. . waves of all 

the medium are small compared to the wave lengths, length0 . 
the velocity becomes the same for waves of all lengths. 












294 


NATURAL PHILOSOPHY. 


This is the case 
with sonorous 
waves; 


This is the case with sonorous waves, Equation (3), 
Acoustics, whose lengths vary from several inches to seve¬ 
ral feet; compared to which distances those between 
the consecutive molecules of air may be regarded as 


insignificant. 


But is not true 
ft>r luminous 
waves: 


Velocity greatost 
for red and least 
for violet waves. 
Kcfractive index 
varies with the 
wavo length; 


Is greatest for 
lavender grey 
and least for red. 
Effect of an 
oblique incidence 
of white light. 


§ 109. In the ethereal medium , whose vibrations pro¬ 
duce light, however, as it exists in the various forms 
of natural bodies, the above conditions do not obtain. 
In the ether of the atmosphere, for example, the lumin¬ 
ous waves, we have seen, vary in length from 0,0000167 
to 0,0000266 of an inch, compared to whicli distances 
those between the adjacent molecules have a sensible 
value; the last factor in Equation (92), cannot, therefore, 
be unity, and the velocity of w^ave propagation must 
depend upon the wave length. A consideration of the 
equation shows that the velocity will be greatest for the 
red and least for the violet waves. 

The index of refraction of any substance is, Equation 
(2), the ratio of the velocity of the luminous wave 
through the ether of a Torricellian vacuum to that 
through the ether of the body. And the relative index 
of two bodies is the ratio of the velocities through their 
respective ethers. Hence, both the absolute and rela¬ 
tive indices vary with the wave lengths, being greatest 
in lavender grey and least in red, those of violet, indigo, 
blue, green, yellow, and orange, lying intermediate be¬ 
tween these. 

When, therefore, the waves which constitute white 
light fall obliquely and simultaneously upon the face of 
a new medium, they will all be deviated on account of 
the change of density and elasticity of the ether which 
they then encounter, and the intromitted waves will be 
unequally deviated, because of their difference of wave 
length; these waves will, hence, separate from each other 
and proceed in different directions; and, if intercepted by 
any reflecting surface, as a screen, will exhibit thereon 
their respective colors. 



ELEMENTS OF OPTICS. 


295 


§ 110. This is well illustrated by the action of an opti- Experimental 

Cal prism. Let a illustration; 

beam $/S\ of solar Fi s* re¬ 

light, be admitted in¬ 
to a dark room 
through a small hole 
in a window shutter, 
and received upon a 
screen X Y\ it will 
exhibit a round lu¬ 
minous spot at T\ in 
the direction of S/S' produced; but if the face of a re¬ 
fracting prism A B C, be interposed, the spot T will dis¬ 
appear, and there will be formed upon the screen 
an elongated image of the sun, variously and beautifully 
colored, beginning with red on the side of the refract- pass through a 
ing angle A , of the prism, and passing in succession pnsm; 
through orange , yellow, green , blue , indigo , and terminat¬ 
ing in violet and lavender grey , making eight in all. 

These colors are not separated by well-defined bounda- Color8prodac<J(1; 
ries, but run imperceptibly into each other; nor are the 
colored spaces of the same length. The following table 
exhibits the relative lengths of these spaces as obtained 
by Sir Isaac Newton with the glass prism used by him, 
and by Fraunhofer, with a prism made of flint glass. 




Newton. 

Fraunhofer. 


Red - 

45 

56 


Orange - 

27 

27 


Yellow - 

48 

27 

Relative lengths 

Green - 

60 

46 

of the colored 

Blue - 

60 

48 

spaces; 

Indigo 1 

40 

47 


Yiolet and lavender grey, 

80 

109 


Total length, 

360 

360 



This property of luminous waves by which they pos- Unequal 
sess different indices of refraction and are deviated refrangiblhtjr • 









29C, 


Solar speetram. 


Effect of 
admitting the 
light through a 
narrow slit; 


Effect varies 
with the light 


Use of these 
lines; 


Number of 
primary colors 
considered; 


NATURAL PHILOSOPHY. 


through different angles for the same angle of incidence, 
is called the unequal ref Tangibility of light / and the 
colored image thence arising is called the solar spectrum . 

§ 111. When the light is admitted through a very 
narrow slit parallel to the refracting edge of the prism, 
and the prism is of pure homogeneous glass and held in 
the position of minimum deviation, § 25, the whole 
spectrum appears mark¬ 
ed by dark and bright 
lines, all parallel to the 
slit, some being broader 
and better defined and 
more conspicuous than others. With an ordinary prism 
of flint glass, the eye distinguishes about twelve ; Fraun¬ 
hofer, with a fine prism of his own glass, distinguished, 
by the aid of a telescope, six hundred. Certain of these 
lines are at unequal intervals, which also differ for dif¬ 
ferent media, though they are of the same order and in 
the same colored spaces. They differ essentially with 
the light employed: the light of the clouds, of the Moon, 
and of Yenus, show them exactly as in the direct light 
of the sun. The bright fixed stars give lines peculiar 
to themselves, as also do electric lights. The light of 
flames shows none, or at least only certain dark intervals 
under peculiar circumstances. These lines furnish the 
means of measuring the refractive indices of different 
media for different colors. 

§ 112. A question often proposed, as to the number of 
primary colors, can only be answered with reference to 
the sense in which it is asked. If it be meant to apply 
to the number of tints distinguishable in the spectrum, 
this will be a matter of individual judgment to different 
eyes. Newton distinguished seven , Sir John Herschel 
eight , Sir David Brewster three ; but perhaps most ob¬ 
servers would admit that it is impossible to fix on any 
definite number, since the light appears to go through 


Fig. 79. 


















ELEMENTS OF OPTICS. 


297 


every possible shade of color, from the deep red to faint colors of tho 
violet or lavender grey. If we understand the question ^loivcT^to 1 
as applying to the number of definite points at each of eight classes, 
which a wave of different length occurs, their number 
must be considered as infinite. These waves resolve 
themselves into eight classes, distinguished by the color 
they excite in the mind, the same color of different shades 
being produced by waves whose lengths vary between 
certain limits. 


113. To find the index of refraction for any one of to find 


Fig. 80. 


refractive index 
for any color; 



these different co¬ 
lors, let A be a re¬ 
fracting prism,made 
of any transparent 
medium; m n , a gra¬ 
duated circle, to the 
centre of which a 
small telescope is 
attached in such a manner that its line of colli- 
mation shall move in a plane parallel to that of the gra¬ 
duated circle, which is held in a position at right angles Exp]anation . 
to the edges of the prism. The telescope, being pro¬ 
vided at its solar focus with a fine wire perpendicular to 
the plane of the circle, is directed to some distant source 
of light, and the reading of the vernier noted. It is 
then directed so as to receive the colored rays from the 
prism, and the reading again noted when the prism is 
turned to the position giving the deviation a minimum. 

"We shall then have 


DDC=S = DCS+DSC 


or neglecting the very small angle subtended by D C r Two prisms with 
the distance of the object, have 

directions. 


8 = DCS, 





298 


NATURAL PHILOSOPHY. 


Mid<iio of which is the difference of the readings; and this in 
Equation (12), will give the value of m. 
refractive index. If the color occupying the middle of the spectrum he 
taken, we shall find the value of m, which answers to 
what is called the mean deviation, and which is the 
same as that given in the table of article 18. 

If a hole be made in the screen, Fig. (78,) at any one 
of the colors, as green, for example, and this color, after 
passing through, be deviated by a second prism P , no 
further separation of the waves will be found to take 
place, but a green image, of the shape and size of the 
hole in the first screen, will be formed upon a second 
screen held behind at G '. 

The colors of the spec- 


Any color 
deviated a second 
time. 


Result of 
reuniting the 
colors of the 
spectrum. 


trum being received, 
each upon a separate 
mirror, may, by vary¬ 
ing the relative position 
of the mirrors, be re¬ 
united, by reflexion, on 
a screen at IF, where 
a white spot will be 
formed as though it 
were illuminated wit 
common light. 


Fig. 81. 




3N OF LIGHT. 


]>• 


cons. 


eft 
eighi , 
servers 
definite nu. 


just been explained, it appears 
stitute white light may be sepa- 
y refraction. The act of such 
ispersion of light, and that pro- 
y which this is performed, is 


»ht to be incident under a very 












ELEMENTS OF OPTICS. 


299 


small angle on any prism, we may replace the sine of the 
angle of incidence and that of refractioh by the arcs, 
and we shall have from Equations (10), (3) and (3)', by 
accenting the refractive index and refracting angle of the 
prism, 

9 + ± = m' (©' + 4-') = m' . a! Equation* 

v J combined; 

and this in Equation (11), by accenting £, gives 

S' = ( 7/1' — 1) . OL ^ .(93) Equation for on* 

prism; 

from which it appears that the deviation will increase 
with the refractive index and refracting angle of the 
prism. 

For a second prism, we have in like manner, 

S" = {m!' — 1). a" same for a 

second prism;] 

and the same for others; and for a number n of prisms 
we have, by taking the sum of all the deviations, and 
denoting the total deviation by 


S l)a' + (m" — 1) a" + (m'"-l)a'". .. (m n — 1) a n . .(94) Same for any 

number. 

in which, as long as the index of refraction exceeds unity, 
the terms will have the same sign when the refracting 
angles of the prisms are turned in the same direction, 
but contrary signs when these angles are turned in oppo¬ 
site directions. 

In the case of two prisms whose refracting angles are 
turned in opposite directions, Equation .(94) becomes 


s 2 =(m'- 1 - 1) a " 


Two prisms wills 
refracting angles 


and if S. be zero, or there be no final deviation, we have in o^ite 

directions. 


(m'-l)a' - (*»"- 1) a"= 0 




300 


NATURAL PHILOSOPHY. 


Condition that 
will produce no 
deviation for a 
particular color. 


To find the 
dispersive power 
of at>y medium; 


Equations for 
the extreme and 
mean colors; 


Reductions; 


Notation; 


or 


m! - 1 _ a" 
m" - 1 ^ a 5 

whence we see, that if the refracting angles of two 
prisms be in the inverse ratio of the excess of the indices 
of refraction of any wave above unity, this wave will 
not be finally deviated by the action of both prisms. 

§ 116. Kesuming Equation 
(94), and denoting the devia¬ 
tion T n V, of the violet, Tn A?, 
of the red, and Tn G, of the 
green waves by £ r ,'£ r , and re¬ 
spectively, the green being 
the mean of the spectrum, we 
have 

= ( m 'v - 1 ) 

8 r = (m' r - l) a',f 
S e =(#*',“ 1) a'; 

in which m' v , m' r , and m’^ denote respectively the in¬ 
dices of refraction of the violet, red, and green waves. 

Subtracting the second from the first, and dividing 
by the third, there will result 


whence, the quotient arising from dividing the angle 
RnV, subtended by the spectrum, by the angle Tn G , 
of mean deviation, is constant for the same medium, and 
is therefore taken as the measure of the dispersive power 
of the medium. And denoting this quotient by D : the 
foregoing Equation gives, omitting the accents, 


m v — m. 

m’ — 1 


. . (95) 


Fig. 82. 













ELEMENTS OF OPTICS. 


301 


J) _ m v - ™r 

m g — 1 


. Y a l ue for 
\tfOj dispersive power. 


By this formula, after finding the values of m v , m r , 
and m g , in the manner indicated, the dispersive powers 
£of the substances named in the following table, as well 
as those of many others, were obtained. 


TABLE OF DISPERSIVE POWERS. 


Substances. 

i 1 

m v — m T 

Realgar melted, 

0,267 

0,394 

Chromate of Lead, 

0,262 

0,388 

Oil of Cassia, 

0,139 

0,089 

Flint Glass, 

0,050 

0,032 

Crown Glass, 

0,033 

0,018 

Olive Oil, 

0,038 

0,018 

Water, 

0,035 

0,012 

Muriatic Acid, 

0,043 

• 0,016 


There is a circumstance connected with this subject 
which has been already alluded to, and which should be 
carefully noticed, owing to its importance in the con¬ 
struction of lenses. If the lengths of spectra formed by 
two prisms of different media be the same, the colored 
spaces in the one will not, in general, be equal in length irrationality of 
to the corresponding spaces of the other. This circum- dis P ersion - 
stance has been called the irrationality of dispersion. 

§117. It is one of the popular, and at first view Objection to 
plausible, objections to the theory, just explained, of the oTthe 
constitution of white light, and especially of the unequal constitution of 
velocities of waves of different lengths, that a star when whlte 1,ght 5 
shut out from view, by the interposition between it and 
the earth of any opaque and non-luminous body, should 
exhibit at its disappearance tints of color due to the suc¬ 
cessive elimination from its light of the red, orange, yel¬ 
low, and so on, in the order of the spectrum, while at 











302 


NATURAL PHILOSOPHY. 


Objection 

answered. 


Conclusion 
drawn from 
known 
principles ; 


And appearances 
thus accounted 
for. 


its reappearance it should present the complementary 
hues of these tints in the reverse order as to time; whereas 
no such phenomena are found to take place. The objec¬ 
tion, however, assumes what we have no right to grant, 
viz.: that the relations of the wave lengths to the 
distances between the adjacent molecules in the great 
atmosphere of ether which connects us with the plane¬ 
tary and stellar regions, are the same as in the ether 
which pervades the bodies that make up the materi¬ 
als of the earth. But we have just seen that the wave 
lengths, as a general rule, diminish as the densities of the 
bodies in whose ether the waves exist, increase, while, 
on the contrary, the distances between the ethereal mole¬ 
cules may increase. 

It would be more consonant to the principles of in¬ 
duction, to adopt the law expressed by Equation (92), 
which is but the simple consequence of known physi¬ 
cal principles, and conclude from the non-appearance ot 
color at the occultation of a star, that the distances be¬ 
tween the ethereal molecules which occupy the celestial 
regions are insignificant in comparison to the wave 
lengths. This would bring the final waves at disappear¬ 
ance, of whatever length, all to the spectator at the 
same instant; and the same being true of the first waves 
at reappearance, there should be no color. 


CHROMATIC ABERRATION. 


Chromatic 
aberration ; 


Illustration; 


§ 118. It fol¬ 
lows from the un¬ 
equal refrangibili- 
ty of the elements 
of white light, 
that the action of 
a lens will be, to 
separate these el- 


Fig. 88. 






ELEMENTS OF OPTICS. 


303 


ements and direct them to different foci, since the value Elements °f 
of/", in Equation (27), depends upon that of m. Sub-1“,°^ 

stituting in that equation — for/ i-LY in the case differe °‘ “ 


P \r 


r 


of a spherical lens ; and writing f v and f r , for the focal 
distances of the violet and red rays, we obtain 



Relation bet ween 
the conjugate 
focal distances 
for red and 
violet: 


in which m.„, being greater than f v , will be less than 
/ r , and the violet rays will be brought to a focus soonest. 

This departure from accurate convergence, caused by the 
unequal refrangibility of the elements of white light, when Chromatlc 
deviated by a lens, is called chromatic aberration , and aberration 
depends upon the nature of the lens and not on its defined; 
figure. It is measured, along the axis of the lens, by the its measure, 
value of f r —/». 

The intersection of the cone of violet rays, with that 
of the red rays, will give what is called the circle of circle of least 
least chromatic aberration. The diameter and position chromatie 

nhorrntinn • 


of this circle can readily be found. From the point 
s , demit the perpendicular s O — y, to the axis ; 
this will divide f r — /<,, into two parts v 0 = x, and 
O r — w \ and calling the radius of aperture of the lens 
a , we shall obtain from the similar triangles of the figure, 


« fr f. 


y __ w x 


Relations from 
the figure; 


whence we deduce 


W+X=f r -f v = y - (./ +./) 

(m 


Same reduced 




304 


NATURAL PHILOSOPHY. 


Radius of circle 
of least 
chromatic 
aberration; 


Diameter of 
the same; 


Measure of 
chromatic 
aberration. 


Same in a 
different form; 


Final value for 
diameter of 
circle of least 
chromatic 
aberration. 


or 


y = a 


fr—f, 

fr + fv 


. . (97) 


The denominator of this expression is* equal to twice 
the mean value of f'\ and therefore, 

« 

2 y = «(/r -/.)• jr,; 

and from Equation (27), we have 

1 1 _ m v — 1 m r — 1 __ m v — m r 

77 fr ~ P P ~ P 


or 


fr-fv = 



■f"\ 


by substituting f' 2 , for f r .f v , to which it is nearly equal. 

Substituting the value of p, from second equation of 
group (30), the above becomes 


fr ~fv = 


/"» 

m-1 ' F„ 


hence, 


2 y = a. 


m v — m r 
m — 1 


r 

'K 



(98) 


In the case of parallel rays, the last factor is unity, 
from which we conclude, that the diameter of the circle 
of least chromatic aberration is equal to the radius of 
aperture of the lens , multiplied by the dispersive power. 

The distance of this circle from the lens is. 











ELEMENTS OF OPTICS. 


305 


fv + X=fv + 


fv-y 


replacing by its value in Equation (97), we have 
a 


Distance of this 
circle from the 
lens; 


f , 2 /„/, fqQ ' i Th9Mm9 

Jv + * — y. .. \ JV ) reducod. 

The effect of chromatic aberration is to give color to Effect of 
the image of an object, and to produce confusion of chromatic 
vision in consequence of the different degrees of conver¬ 
gence in the differently colored waves proceeding from 
the same point of an object. The vertices of the cones 
composed of the rays of these waves, lying in the axis, 
every section perpendicular to this line will have its 
brightest point in the centre, and the yellow waves con- in part 
verging nearly to the mean focus, and having by far the destr03red; 
greatest illuminating property, the bad effects which 
would otherwise arise from this aberration are in part 
destroyed. Besides, these effects may be lessened by 
reducing the aperture of the lens, though not in the May be 
same degree as those arising from spherical aberration, diminished. 


ACHROMATISM. 

§ 119. It is, then, impossible, by the use of a -single Achromatism; 
homogeneous lens, to deviate the different waves of white 
light accurately to a single focus, and, consequently, im¬ 
possible, by the use of such a lens, to form a. colorless 
image of any object; both, however, may be done by the 
union of two or more lenses of different dispersive powers. 

The principle according to which this maybe accomplished, Achromatio 
is termed Achromatism , and the combination is said to combinatlon *. 
be achromatic. 

Let us suppose two lenses of different dispersive powers 
placed close together. The focus of the combination will, 

20 






304 


NATURAL PHILOSOPHY. 


Radius of circle 
of least 
chromatic 
aberration; 


Diameter of 
the same; 


Measure of 
chromatic 
aberration. 


Same in a 
different form; 


Final value for 
diameter of 
circle of least 
chromatic 
aberration. 


or 


y = a 


fr—fv 

fr + fv 


. . (97) 


The denominator of this expression is^ equal to twice 
the mean value of f'\ and therefore, 


2 y = a(fr -/,) • JT ,; 

and from Equation (27), we have 


1 1 _ m v — l m r — l_m v — m r 

77 fr ~ ? f ~ f 


or 


jf _y» _ m r y"2 


by substituting f" 2 , for f r .f v , to which it is nearly equal. 

Substituting the value of p, from second equation of 
group (30), the above becomes 


f s _ ~ m r /"» 

Jt Jv ~ m — 1 ' F„ ' 


hence, 


/" _ 


/" 


2 y = a. Vh- _. -L — — a. D. I _ 

m — 1 F„ F„ 


■ (98) 


In the case of parallel rays, the last factor is unity, 
from which we conclude, that the diameter of the circle 
of least chromatic aberration is equal to the radius of 
aperture of the lens , multiplied by the dispersive power. 

The distance of this circle from the lens is. 











ELEMENTS OF OPTICS. 


305 


Distance of this 
circle from the 
lens; 

replacing fL by its value in Equation (97), we have 


fv + » =/„ + 


fv y 


fv + » = 


2 fvfr 
fr +fv ' 


. . (99) 


The same 
reducod. 


The effect of chromatic aberration, is to give color to Effect of 
the image of an object, and to produce confusion of chromatic 
vision in consequence of the different degrees of conver¬ 
gence in the differently colored waves proceeding from 
the same point of an object. The vertices of the cones 
composed of the rays of these waves, lying in the axis, 
every section perpendicular to this line will have its 
brightest point in the centre, and the yellow waves con- in part 
verging nearly to the mean focus, and having by far the destroyed; 
greatest illuminating property, the bad effects which 
would otherwise arise from this aberration are in part 
destroyed. Besides, these effects may be lessened by 
reducing the aperture of the lens, though not in the May be 
same degree as those arising from spherical aberration, diminished. 


ACHROMATISM. 

§ 119. It is, then, impossible, by the use of a’single Achromatism ; 
homogeneous lens, to deviate the different waves of white 
light accurately to a single focus, and, consequently, im¬ 
possible, by the use of such a lens, to form a. colorless 
image of any object; both, however, may be done by the 
union of two or more lenses of different dispersive powers. 

The principle according to which this maybe accomplished, Achromatic 
is termed Achromatism , and the combination is said to combinatlon -. 
be achromatic. 

Let us suppose two lenses of different dispersive powers 
placed close together. The focus of the combination will, 

20 





306 


NATURAL PHILOSOPHY. 


Two lensea 
taken ; 


Focus for red; 


Focus for violet; 


X 

Equating these 
focal distances; 


Relation 
obtained ; 


Same in a 
different form; 


Explanation of 
the result; 


Equation (34), and the fourth Equation of group (30), for 
any one of the elementary colors as red, be given by 

1 m r — l mj — 1 1 . 

+T . 

and for violet, 

1 m v — 1 , v/ V ~ 1 , 1 . 

fv P P / 


If f" and f v '\ were equal, the chromatic aberration, 
as regards these colors, would be destroyed; equating 
them we have, 

(m T - 1) P '+ (m/ - 1) P = (m v - 1) p' + (m v ’ - 1) p 


whence, 

JL = ~~ 1 ) ~ ( yw * - 1 ) = _ m v ~ 

?' {m'r- l)-(m/~l) m v r - ra r ' 


the second member being negative, because m'„, is greater 
than m' r . 

Multiplying both members of this equation by- 

it may be put under the form, 


m' — 1 m v — m r 

p' m — 1 

m —-1 m' v — m' r 

P m' — 1 


( 100 ) 


The second member expresses the ratio of the disper¬ 
sive powers of the media, and the first, the inverse ratio 
of the powers of the lenses for the mean waves; this 
being negative, one of the lenses must be concave, the 
other convex; and the powers of the lenses being inversely 
















ELEMENTS OF OPTICS. 


307 


as the focal distances, we conclude, that chromatic aber- Kau> for 
ration, as regards red and violet , may be destroyed by achromatic^ 
uniting a concave with a convex lens, the principal focal for TQd 
lengths being taken in the ratio of their dispersive powers . V10let ’ 

The usual practice is to unite a convex lens of crown 
glass with a concave lens of flint glass, the focal distance usual 
of the first being to that of the second as 33 to 50, combinatioa * 
these numbers expressing the relative dispersive pow- • 

ers as determined by experiment; (see Table § 116). The 
convex lens should have the greater power, and, there¬ 
fore, be constructed of the crown-glass; otherwise, the 
effect of the combination would be the same as that of convex icns 
a concave lens with which it is impossible to form a 6hon,d have th0 

. # greater power; 

real image of a real object. 


Fig. 84. 



Illustration; 


To illustrate : let parallel rays be received by the lens 
A ; its action alone would be, to spread the different 
colors over the space VB, whose central point m, is dis¬ 
tant from A, 33 units of measure, (say inches), the violet 
being at Y, and the red at B ; the action of the lens B, 
alone would be, to disperse the rays as though they pro¬ 
ceeded from different points of the line V’ B', whose Explanation of 
central point m', is distant from B, 50 inches, the violet the action f tho 
appearing to proceed from V', and the red from B r \ and the 
effect of their united action would be, to concentrate the 
red and violet at F, whose distance from the lens is 
equal to the value of F, deduced from the formula 



1 

97,06 


inches. 


Example; 







308 


NATURAL PHILOSOPHY. 


Po nt in which 
red and violet 
WDuld be united; 


Geometrical 
illustration; 


Why the other 
colors would not 
generally be 
concentrated in 
the same point; 


Secondary 

spectrum. 


Substances 
which fulfil the 
conditions for 
perfect 

achromaticity; 


or 


F — — 97,06 inches. 


Fig. 84 





Now, any one of the colors, orange for example, at 
(?, in the space B V, which is thrown by the 
convex lens in advance of the centre m, and the same 
color at O’ in the space V' B\ which is thrown by the 
concave lens behind the centre m', will, it is obvious, be 
united with the violet and red at F\ by the joint action 
of both lenses ; and the same would be true of any other 
color, but for the irrationality of dispersion of the me¬ 
dia of which these lenses are composed, which prevents it, 
and hence an image formed by such a combination of 
lenses will be fringed with color; the colors of the fringe 
constituting what is called a secondary spectrum . An 
additional lens is sometimes introduced to complete the 
achromaticity of this arrangement. 

§ 120. If two lenses, constructed of media between* 
which there is no irrationality of dispersion, be united 
according to the conditions of Equation (100), the com¬ 
bination will be perfectly achromatic. It is found that 
between a certain mixture of muriate of antimony with 
muriatic acid, and crown-glass, and between crown-glass 
and mercury in a solution of sal ammoniac, there is lib 
tie or no irrationality of dispersion. These substances 
have therefore been used in the construction of com¬ 
pound lenses which are perfectly achromatic. The figure 







ELEMENTS OF OPTICS. 


309 


represents a section of one of these, Fi & ^ Representation 

consisting of two double convex 
lenses of crown-glass, holding be¬ 
tween them, by means of a glass 
cylinder, a solution of the muriate in 
the shape of a double concave lens, 
the whole combined agreeably to the relations expressed 
by Equation (100). The focal distance of the convex 
lenses is determined from Equation (31). 

§121. From Equation (98), we infer, that the circle circle of least 
of least chromatic aberration is independent of the focal c J ,romaUc 

A aberration 

length of the lens, and will be constant, provided the independent or 
aperture be not changed. Now, by increasing the focal focal length: 
length of the object glass of any telescope, the eye lens 
remaining the same, the image is magnified; it follows, 
therefore, that by increasing the focal length of the field 
lens, we may obtain an image so much enlarged that 
the color will almost disappear in comparison. Besides, Telescopes 
an increase of focal length is attended with a diminution former, y veiy 
of the spherical- aberration. This explains why, when 
single lenses only were used as field lenses, they were 
of such enormous focal length, some of them being as 
much as a hundred to a hundred and fifty feet. The 
use of achromatic combinations has rendered such lengths 
unnecessary, and reduced to convenient limits, instru-Modem ones 
ments of much greater power than any formerly made 8hortcr ' 
with single lenses. 



INTERNAL REFLEXION. 

§ 122. Whenever the w r aves of light in their motion 

° Internal 

through any medium meet with a change of density and reflexion; 
elasticity, they will be both reflected and refracted. In 
consequence, objects seen by reflexion from a plate of 







310 


NATURAL PHILOSOPHY. 


When objects 
seen by reflexion 
from glass 
appear double; 


Relative 

brightness of the 
images when the 
second surface is 
in contact with 
various 
substances; 


glass, in the atmosphere, appear double when the faces of 
the glass are not parallel, there being an image formed 
by reflexion from each face. The image from the second 
surface will be brighter in proportion as the obliquity or 
angle of incidence of the incident waves becomes greater. 
If the second surface of the glass be placed in contact 
with water, the brightness of the image from that surface 
will be diminished; if olive oil be substituted for the 
water, the diminution will be greater, and if the oil be 
replaced by pitch, softened by heat to produce accurate 
contact, the image will disappear. If, now, the contact 
be made with oil of cassia, the image will be restored ; 
if with sulphur, the image will be brighter than with oil 
of cassia, and if with mercury or an amalgam, as in the 
common looking-glass, still brighter, much more so indeed 
than the image from the first surface. 

The mean refractive indices of these substances are as 
follows : 


Refractive 
indices of these 
substances; 


Air, ----- 1,0002 
Water, - - - - 1,336 

Olive Oil, - - - 1,470 
Pitch, - - 1,531 to 1,586 
Plate Glass, 1,514 to 1,583 
Oil of Cassia, - - 1,641 
Sulphur, - - - 2,148 


Indices 

compared with 
the index for 
plate glass; 


Conclusion. 


Taking the differences between the index of refraction for 
plate glass and those for the other substances of the 
table, and comparing these differences with the forego¬ 
ing statement, we are made acquainted with the fact, 
which is found to be general, viz.: that when two media 
are in perfect contact, the intensity of the light reflect¬ 
ed at their common surface will be less, the nearer their 
refractive indices approach to equality; and when these 
are exactly equal, reflexion will cease altogether. This 
is an obvious consequence of the rationale of reflexion, 
given in Acoustics, § ^ 



ELEMENTS OF OPTICS. 


311 


§123. Different substances, we have seen, have, in° win gtoa 
general, different dispersive powers. Two media may, 
therefore, be placed in contact, for each of which the power the light 
same color, as red, for example, may have the same in- ^Lmitt^ut 
dex of refraction, while for the other elements of white the second 
light, the indices may be different; when this is the surface ' 
case, according to what has just been said, the red would 
be wholly transmitted, while portions of the other colors 
would be reflected and impart to the image from the 
second surface the hue of the reflected beam; and this 
would always occur, unless the media in contact pos¬ 
sessed the same refractive and dispersive powers. 


ABSORPTION OF LIGHT. 


§ 124. The waves of light which enter any body are Absorption of 
not transmitted without diminution; but in consequence of llght; 
a want of perfect elasticity due to the reciprocal action 
of the molecules of the ether and the particles of the 
body, and owing to the absence of perfect contact of 
the elements of bodies, these waves undergo a series of 
internal reflexions which give rise, as in the case of n 0 w produced, 
sound, to interferences and consequent loss of intensity. 

This action of bodies upon light is called absorption. 

The quantity absorbed is found to vary not only from Qnantity 
one medium to another, but also in the same medium absorbed varies; 
for different colors ; this will appear by viewing the pris¬ 
matic spectrum through a plate of almost any transpa¬ 
rent, colored medium, such as a piece of smalt blue glass , 
when the relative intensity of the colors will appear al¬ 
tered, some colors being almost wholly transmitted, while 
others will disappear or become very faint. Each color 
may, therefore, be said to have, with respect to every 
medium, its peculiar index of transparency as well as of 
refraction. 



312 


NATURAL PHILOSOPHY. 


Quantity 
absorbed 
depends upon; 


Extreme colors 

transmitted 

longest 

Herschel’s 
hypothesis to 
account for the 
extinction of a 
homogeneous 
wave; 


The quantity of each color transmitted, is found to 
depend, in a remarkable degree, upon the thickness of the 
medium ; for, if the glass just referred to be extremely 
thin, all the colors are seen; but if the thickness be about 
2 V of an inch, the spectrum will appear in detached 
portions, separated by broad and perfectly black inter¬ 
vals, the rays corresponding to these intervals being to¬ 
tally absorbed. If the thickness be diminished, the dark 
spaces will be partially illuminated; but if the thickness 
be increased, all the colors between the extreme red and 
vidlet will disappear. 

Sir John Herschel conceived that the simplest 
hypothesis with regard to the extinction of a wave of 
homogeneous light, passing through a homogeneous me¬ 
dium is, that for every equal thickness of the medium 
traversed, an equal aliquot part of the intensity which 
up to that time had escaped absorption, is extinguished. 


That is, if the -^-th part of the whole intensity, which 
m 

will be called c, of any homogeneous wave which en¬ 
ters a medium, be absorbed on passing through a thick¬ 
ness unity, there will remain, 


Fortton n m—7l 

transmitted C ■— C = -C , 

through a unit of ™ ^ 

thickness; 

and if the -^-th part of this remainder be absorbed in 
m 

passing through the next unit of thickness, there will 
remain 


Portion 
transmitted 
through two 
units; 


m — n 
m 


g — 


n (m — n) c _ m — n 




and through the third unit, 



n{m — n)' 



Through three 
units; 


m 












ELEMENTS OF OPTICS. 


313 


and through the whole thickness denoted by t units, 



Through t units 
of thickness; 


So that, calling c the intensity of the extreme red 
waves in white light, c' that of the next degree of re- 
frangibility, c" that of the next, and so on, the incident 
light will, according to Sir J. H., be represented in in¬ 
tensity by 

C + <?' + c" -f O'" + &C. * Intensity of 

incident light; 

and the intensity of the transmitted light, after travers¬ 
ing a thickness £, by 

c f + c r y rt + c" y” 1 + &c. . . . (101) Thatof 

transmitted 

_ light; 

Wherein y, represents the fraction — _which will 

m 

depend upon the waves and the medium, and will, of 
course, vary from one term to another. 

From this it is obvious, that total extinction will be Total €Xtinction 
impossible for any medium of finite thickness; but if finite 6 
the fraction y, be small, then a moderate thickness, which thickness ; 
enters as an exponent, will reduce the fraction to a value 
perfectly insensible. 

Numerical values of the fractions y, y\ y ", &c., may indices of 
be called the indices of transparency of the different transparency ' 
waves for the medium in question. 

There is no body in nature perfectly transparent, though Nobody in 
all are more or less so. Gold, one of the densest of me- trans^arentT^ 
tals, may be beaten out so thin as to admit the passage 
of light through it: the most opaque of bodies, charcoal, 
becomes one of the most beautifully transparent under 
a different state of aggregation, as in the diamond, “ and 
all colored bodies, however deep their hues and however 
seemingly opaque, must necessarily be rendered visible 
by waves which have entered their surface; for if re¬ 
flected at their surfaces they would all appear white 






316 


NATURAL PHILOSOPHY. 


Illustration for 
primary bow; 


flotation; 


Equation for 
one internal 
reflexion ; 


For two internal 
reflexions; 


For n internal 
reflexions; 


Angle subtended 
by the equal 
chords; 


Substitution; 


. Fig. 87. 



refracted, D B the reflected, and B B the emergent raj. 
Call the angle OAm= the angle of incidence, 9 , and 
the angle C AD = the angle of refraction, 9 '; the an¬ 
gles subtended by the equal chords A D and D B, 
and the angle A CB , 6. Then we shall have 

d = — 2 %; 

and if there be two internal reflexions, there will be 
three equal chords, in which case, 

d = 2 3x; 

and generally, for n internal reflexions, 

d = 2 * - n + 1 .x.( 102 ) 

but in each of the triangles whose bases are the equal 
chords, and common vertex the centre of the drop, 

X = — 2 <p' 

and this, in Equation ( 102 ), gives, on reduction, 

d=2(^ + l)9'-(^vl)^ • • • .(103) 

Because the chords are all equal, the last angle of in¬ 
cidence CB Z>, within the drop in Fig. (87), or C B D\ 




317 


ELEMENTS OF OPTICS. 


in Fig. ( 88 ), is equal to the angle of refraction C A D, Ang!e of 

° 77 A 0 . 7 emergence equal 

and hence the angle of emergence C B m\ is equal to to angle of 
the angle of incidence C A m. incidence 

The angle A 0 B, in Fig. (87), is the supplement of the 
total deviation of the emergent from the incident raj, 

^and is equal to the angle B EF, subtended bj the ra- References to 
^dius of the bow; in Fig. ( 88 ), it is the excess of total de- fisures; 
viation above 180°. 


Calling this angle £, we shall have 


Notation and 
equation; 


s = =p{2(p — 6); 

Fig. 88. 



Illustration for 
secondary bow; 


the upper sign referring to Fig. (87), and the lower to 
Fig. ( 88 ); replacing d, bj its value in Equation (103,) the 
above reduces to 


General value 

& = ^(2<p-2(n+l)<p' .«) . (104) fOT ra,Ji " s of a 

‘ ' ' 7 colored arch; 


this, with equation 


sin 9 = m . sin 9 ', 


. (105) 


From which the 
radius of any 
particular color 


will enable us to determine the value of 5, when 9 and m ^ be found 


are given for any particular color. 

For any value of 9 , assumed arbitrarily, 8 will, in gen¬ 
eral, correspond to rays of the same color so much dif¬ 
fused as to produce little or no impression upon the eye; 
but if 9 be taken such as to give 8 a maximum or mini- 




318 


NATURAL PHILOSOPHY. 


What waves 
appertain to the 
rainbow. 


Eelation that 
will fulfil the 
conditions for 
color; 


Corresponding 
angle of 
incidence: 


bame for one 
internal 
reflexion ; 


Eadii of the 
colors of 
primary bow 
deduced; 


Example, red of 
the primary; 


mum,. then will the rays corresponding to m, emerge pa¬ 
rallel, or nearly so, for a small variation in the angle <p 
on either side of that from which this maximum or 
minimum value of 8 results; hence, the waves which en¬ 
ter the eye in this case will be sufficiently copious to 
produce the impression of color, and these are the waves 
that appertain to the rainbow. 

§ 126. By an easy process of the calculus it is found 
that the relation which will satisfy these conditions, is 

1 cos 9 * 
n+ 1 m cos cp' 

Clearing the fraction, squaring both members, adding 
m 2 sin 3 9 ' = sin 3 <p 


and reducing, we get 


cos 9 


V 7 : 


m 3 — 1 


n 2 + 2 n 


(106) 


For one internal reflexion, which answers to Fig. (87), 


cos 9 —a 1 . 

V 3 ’ 

I 

and substituting in succession the values of m, answering 
to the different colors for water, we shall have values for 
9 , and consequently for 9 ', Equation (105), which substi¬ 
tuted in Equation (104), will give the angles subtended 
by the radii of the colored arches which make up what 
is called the primary bow. 


For red, m = 1,3333, hence 

cos 9 = 0,5092 = cos 59° 21', 
sin 9 = VI —cos 3 9 = 0,8603 ; 

See Appendix No. S. 










ELEMENTS OF OPTICS. 


319 


this last, in Equation (105), gives substitution and 

reduction; 

9 ' = 40° 11', 


and these values of 9 and 9 ', in Equation (104), give 

£ = - 118° 42' + 160° 44' = 42° 02'. value of 


For the violet, m — 1,3456, 

Example, violet 

COS 9 = 0,5199 = COS 58° 41V, of the primary; 

sin 9 = 0,8543, 

9 ' = 39° 25', 

S' = - 117° 23' 4-157° 40' = 40° 17'; value of S '; 

hence, the width of the primary bow is 


S - S' = 42° 02' - 40° 17' = 1° 45'. 


Width of primary 
bow. 


If there be two internal reflexions, as in Fig. ( 88 ), we 
shall, by making n — 2 , find 


cos 9 



Solution for two 
internal 
reflexions; 


and obtain, by a process entirely similar, the elements secondary t»w ; 
of what is called the secondary lore. 

For the red rays, 


S = 50° 57', 


Value of S » 


violet, 


(5' = 54° 07', 


Value of O'; 


and 


ff-S = 3° 10', 


Width of 
eocondarv how; 





320 


NATURAL PHILOSOPHY. 


Arrangement of 
the colors in the 
two bows: 


Space between 
them; 


When these 
bows will be 
invisible; 

To find elements 
of a tertiary bow; 


Tertiary not 


the value of 5' in the secondary, being greater than 
the violet will occupy the outside, and the colors, there¬ 
fore, be arranged in an order the reverse of that in the 
primary. Taking the difference between the values of 
S in the primary and secondary bows, we will obtain the 
space between them, which is 50° 57'— 42° 02' = 8° 55'. 
The solar disk being about 32', the width of both bows 
must be increased by this quantity, the solution having 
been made upon the supposition that the light flows from 
a point. The primary is, therefore, 2° 17' in width, and 
the secondary 3° 42'. The half of 32' being added to 
the radius of the red in the primary, will give 42° 18', 
hence, if the sun be more than that height above the 
horizon, this bow cannot be seen. When higher than 
54° 23', no part of the secondary will be visible. 

By substituting in Equation (106), 3 for n, we might 
find the radii of a third bow, which would be found to 
encircle the sun at the distance of about 43° 50'; but the 
proximity of the sun, together with the great loss of 
light arising from so many reflexions, renders this bow 
so faint as to produce no impression; it is, therefore, 
never seen. 


Fig. 89. 



a maximum §127. By means of the calculus it is easily shown 
InVminimum 7 ^at ^ in Equation (104), is a maximum for the primary 
for the secondary; and a minimum for the secondary. This explains the 













ELEMENTS OF OPTICS. 


321 


remarkable fact that the space between these bows always Eemarkabl ° 
appears darker than any other part of the heavens in the the heavens 
vicinity of the bow ; for, no light twice refracted and once between the80 
reflected can reach the eye till the drops arrive at the prim- for. 
ary, and none which is twice refracted and twice reflected, 
can arrive at the eye after the drops pass the secondary ; 
hence, while the drops are descending* in the space be¬ 
tween the bows, the light twice refracted with one or two 
intermediate reflexions, will pass, the first above, and the 
second below or in front of the observer. 

The same discussion will, of course, apply to the lunar 
rainbow which is sometimes seen. 


§128. Luminous and colored rings, called halos , are Hal0S: 
occasionally seen about the sun and moon; the most re¬ 
markable of these are generally at distances of about 
twenty-two and forty-five degrees from these luminaries, 
and may be accounted for upon the principle of unequal 
refrangibility of light. They most commonly occur in 
cold climates. It is known that ice crystallizes in minute 
prisms, having angles of 60° and sometimes 90° ; these The5r a PP earance 
floating in the atmosphere constitute a kind of mist, and 
having their axes in all possible directions, a number 
will always be found perpendicular to each plane pass¬ 
ing through the sun or moon, and the eye of the obser¬ 
ver. One of these planes is indicated in the Figure. 
jS m, being a 

beam of light pa- Fig.. 90 . 

rallel to S E\ 

drawn through the 
sun and the eye, 
and incident upon 
the face of a prism 
whose refracting 
angle is 90° or 60°, 

we shall have the value of £, corresponding to a minimum 
from Equation (12), by substituting the proper values of 
m for ice. The mean value being 1,31, w r e have 
21 









322 


NATURAL PHILOSOPHY. 


Example firat, 


sini(S+ 60°) = 1,31. sin 30° 

\ 8 = 40° 55' 10" — 30° = 10° 55' 10" 
5 = 21° 5Q'20" 


and 


BujKnple second ; 


sin i (5 + 90°) = 1,31. sin 45° 

15 = 67° 52' — 45° = 22° 52' 
5 = 45° 44'. 


Other phenomena of a similar nature will be noticed 
hereafter. 


POLARIZATION OF LIGHT. 


Retrospective 
view of the 
phenomena of 
unpolarized 
light 


Remarks on the 
disturbance of 
molecular 
equilibrium; 


§ 129. We have thus far been concerned with the pro¬ 
pagation of luminous waves through homogeneous media, 
wdth the deviation which these waves undergo on meet¬ 
ing with a change of density, and with the superposi¬ 
tion of two or more waves, by which their effects are 
increased, diminished, or totally destroyed. We now 
come to a class of optical phenomena whose explanation 
depends upon considerations affecting the particular mode 
of molecular vibrations in these waves. 

When an ethereal molecule is displaced from its posi¬ 
tion of equilibrium, the forces of the neighboring mole¬ 
cules are no longer balanced, and their resultant tends 
to drive the displaced molecule back to its position of 
rest. The displacement being supposed very small in 
comparison with the distance between the molecules, the 
forces thus excited will, we have seen in Acoustics, be 



ELEMENTS OF OPTICS. 


323 


proportional to tlie displacement; and according to prin¬ 
ciples explained in Mechanics, the trajectory described by 
the molecule will be an ellipse whose centre coincides 
with the position of equilibrium. Hence, the vibration A distarbed 

, . til.. molecule in 

ot the ethereal molecules is, m general, elliptic, and the general,doscribea 
nature of the light thence arising depends upon the re- an elli P se ? 
lative directions and magnitudes of the axes. These el¬ 
liptic vibrations are in planes parallel to the wave front, 
and consequently transverse to the direction of wave pro¬ 
pagation. The axes of the ellipses may either preserve 
constantly the same direction in their respective planes, 
or may be continually shifting. In the former case the light Distinction 
is said to be polarized; in the latter, it is unpolarized p 0]arize( i and 

Or Common light. common light 

§ 130. The relative magnitude of the axes of the ellip- Nature of thG 
ses determines the nature of the polarization. When determined ; 

the axes are equal the ellipses become circles, and the 
light is said be circularly polarized, when the lesser axis 
vanishes, the ellipse becomes a right line, and the light 
is said to b z plane polarized —the vibrations being in this 
case confined to a single plane passing normally through 
the wave front. In intermediate cases the polarization 
is called elliptical , and its character may vary indefi-^^®^ 6 ’ 
nitely between the two extremes of plane and circular polarization, 
polarization. 

The term polarization in optics has come to be a misno- Use of the term 
mer. It was introduced before the theory of luminous expiXe ^ 11 
undulations had gained much favor with the scientific 
world; and was intended at the time of its adoption to 
express certain fancied affections, analogous to the polari¬ 
ties of a magnet, conceived to exist in the material ema¬ 
nations which, according to Newton, constituted the es¬ 
sence of light. It would be better were it replaced by 
some other term more expressive of the actual condition 
of the light; but at present this seems to be impossible, 
owing to its very general acceptation, and it is accord¬ 
ingly retained. 



324 


NATURAL PHILOSOPHY. 


Illustration by a § 131. To conceive the manner in which an undulation 
stretched cord; ma y propagated by transversal vibrations, imagine a 
cord stretched horizontally, one end being attached to 
a fixed point and the other held in the hand. If the lat¬ 
ter extremity be made to vibrate by moving the hand 
up and down, each particle of the cord will, in succes¬ 
sion, be thrown into a similar state of vibration, and a 
series of waves will be propagated along the cord with 
a constant velocity. The vibrations of each succeeding 
particle of the cord being similar to that of the first, 
will all be performed in the same plane, and the whole 
ethwed particles represent the state of the ethereal particles along a 
in a plane plane polarized wave. The plane of vibration is called 

polarized wave; pl ane of polarization. 

If, after a certain number of vibrations in the vertical 
plane, the extremity of the cord be made to vibrate in 
some other plane, and then in another,—and so on in 
rapid succession—each particle of the cord will, after a 
certain time, proportional to its distance from the hand, 
assume in succession all these varied vibrations ; and the 
whole cord instead of taking the form of a curve lying 
in one plcme, will be thrown into a species of helical curve , 
depending on the nature of the original disturbance, 
condition of the g uc ] 1 j s the condition of the ethereal molecules in waves 

ethereal particles 

in common light, of common or unpolarized light. 

'undulation When, therefore, we admit a connection among the 
LaTsversai b7 molecules of ether, similar to that which exists among 
vibrations. the particles of the cord, there is no difficulty in con¬ 
ceiving how a vibration may be propagated in a dilec¬ 
tion perpendicular to that in which it is executed. The 
particles of ether, it is true, are not held together by 
cohesive forces like those of a cord, but the molecular 
forces which subsist among them, are of the same kind, 
and produce similar effects. Neither the particles of 
the cord nor the ethereal molecules are in contact. 

§ 132. These illustrations being understood, conceive a 
transversal vibration to proceed from a disturbed mole- 



ELEMENTS OF OPTICS. 


325 


cule at A, towards C, and 
suppose the vibration to take 
place in the plane of the pa¬ 
per, and let M JV be the front 
of the wave at the expiration 
of any time £, after the be¬ 
ginning of motion. The dis¬ 
placement a?, of the molecule 
at (7, will, § 55, Acoustics, be 


Fig. 91. 



given by the equation 


Transversal 
vibration 
supposed to 
proceed from a 
disturbed particle 
at A] 



sin 12 * . 


Vt - G 

X 


Consequent 
displacement of 
another particlo 
at C; 


in which a denotes the amplitude of the disturbance at 
unit’s distance from A ; <?, the distance from A to C\ V, the 
velocity of wave propagation, and X the length of the wave. 

At the same instant, suppose a second transversal vibra- A 8econ(1 
tion to proceed from any other point, as B, towards <7, vibration from 
the vibrations in the latter case being perpendicular to * n y other P oin t; 
the former, and let M' N’ be the front of the wave at 
the expiration of the same time £, as above. The dis¬ 
placement y, of the molecule C, due to this action, will 
be given by the equation 



. sin ^2 . 



Displacement of 
the same particle 
at C\ 


in which h denotes the amplitude of the disturbance at 
unit’s distance from B, and c / the distance from B to G. 

Dividing these equations respectively by the coefficient 
of the circular function in the second member, we obtain 
the equations, 


Vt-c 

. ,cx 


X 

= sin" 1 —, 
a 

Equations 
obtained from 

these 

displacements; 

Vt — c, 

• i C /V 

= sin -1 ; 


X 

b ’ 









326 


NATUPwAL PHILOSOPHY - . 


Cosines of these 
arcs; 


the cosines of these arcs will be respectively 


\/l-^’aiid \/l- 


c?y 2 


Subtracting the second of these equations from the first, 
we find, 


Combining these 
equations and 
reducing; 



— sin -1 


cx 

-sin 

a 


-i 


b * 


Taking the cosine of each member of the equation, and 
recollecting that the cosine of the difference of two arcs 
is equal to the sum of the rectangles of the cosines and 
sines, we find, after a slight reduction, 


We obtain the 
equation of an 
ellipse; 


C -±~y 2 + — x 2 _2 cos— (c — c). —x. ^-?/=sin 2 — (c — c)(107) 

b 2 a 2 A a b X 


which is an equation of an ellipse referred to its centre. 
Light eiiipticaiiy The axes of the ellipses in this case preserving the same 
polarized, direction, the light will, from what we have already said, 
be ellijptically polarized, and is obviously compounded of 
two waves plane polarized in planes at right angles to 
each other. 

When 


Supposition In Cl — C=z—, and — = 

Equation (107); 1 4 7 a 5 * 


Equation (107), reduces to 


Which reduces it 
to the equation of 
a circle; 


x 2 + y 2 = a *; . . . . . (108) 


the equation of a circle, and the light becomes circularly 
polarized, being compounded of two plane polarized waves 
Light circularly e( l ua ^ intensity, having their planes of polarization at 
polarized; right angles to each other. In this latter case, the light 








ELEMENTS OF OPTICS. 


327 


will possess many of the properties of common light, but common light 
will differ from it in some important particulars to be ^ a Jompoand?d d 
noticed presently. of two waves 

polarized in 
planes 

§ 133. The difference Fi6 ’ 92 ‘ perpendicular to 

each other; 

common light being, 
that in the former, the 
axes of the ellipses de¬ 
scribed by the molecules remain parallel, while in the 
latter they are incessantly changing their directions; 
common light, like elliptically polarized light, may be 
regarded as compounded of two plane polarized waves, 
of which the planes of polarization are at right angles 
to each other. When these component vibrations are 
separated, each component becomes plane polarized light, separating these 
This separation may be effected, either by causing these component 
component vibrations to take different directions byor- waves ‘ 
dinary reflexion and refraction, by the retardation or ac¬ 
celeration of one over the other, as in the case of double ^ *■ 

3 # Different ways of 

refraction, soon to be explained, or by absorbing one and causing this 
permitting the other to pass unobstructed, separation. 



between polarized and 


POLARIZATION BY REFLEXION AND REFRACTION. 


_ . _ . T „ Polarization by 

§ 134. It is ascertained that when a wave ot common reflexion and by 
light is incident on any transparent medium of uniform refraction; 
density, under a certain angle of incidence, called the 
polarizing angle , the resolution above referred to, takes 
place; the reflected and refracted waves become jDlane 
polarized, the former in the plane of reflexion, and the 
latter in a plane at right angles to it. Both waves lose 
almost entirely the power of being again reflected or re¬ 
fracted when the surface of a second deviating medium introductory 
is presented to either in a particular manner. remarks; 








328 


NATURAL PHILOSOPHY. 


Fig. 93. 


Experimental 
illustration ; 



Explanation of Thus, MN and M'N\ representing two plates of 
glass, mounted upon swing frames, attached to two tubes 
A and B , which move freely one within the other about 
a common axis, let the beam#./}, of homogeneous light, 
be received upon the first under an angle of incidence 
equal to 56° ; reflexion and refraction will take place ac¬ 
cording to the ordinary law, and if the reflected beam 


JD D’, which is sup¬ 
posed to coincide with 
the common axis of 
the tubes, be incident 

Appearance 

when the upon the second re- 

anaiyzer is flector under the same 

perpendicular to . 

the plane of first angle of incidence, 

reflexion; reflector being per¬ 

pendicular to the 
plane of first reflex¬ 
ion, it will be totally 
reflected, there being 
■none refracted. 

But if the tube B, 

The same when be turned about its 

revolved through aX1S > tbe A b eing 

any angle less at rest, the angle of 

incidence on the glass 
M r JV', will remain 
unchanged, refraction 


Fig. 94. 


















ELEMENTS OF OPTICS. 


329 


will begin, and the re¬ 
fracted portion will in¬ 
crease while the reflected 
portion will diminish, till 
the tube B has been turn¬ 
ed through an angle equal 
to 90°, as indicated by the 
graduated circle (7, on the 
tube A ; in which position of the reflector, the beam will 
be totally refracted. Continuing to turn the tube B , the 
reflexion from M' N' will increase, and the refraction 
will decrease, till the angle is equal to 180°, when the 
plane of the first reflexion will be again perpendicular 
to M r and the whole beam will be reflected : beyond A PP earances 

7 , J when the 

this, reflexion will again diminish, and refraction increase, analyzer is 
till the angle becomes 270°, when the beam will be to- revolved thr0Hgh 

° t the other three 

tally refracted; after passing this point, the same phe- quadrants, 
nomena will recur, and in the same order, as in the 
second quadrant, till the tube is revolved through 360°, 
when the restoration of the reflected wave will be com¬ 
plete. The same phenomena would have occurred had Same phenomena 
the second reflector been presented to the refracted com- refrlcted wave.° 
ponent of the original incident wave on its emergence 
from the first plate of glass. 

It is important to remark in this connection, that the 
molecular vibrations in the wave reflected from, and 
in that transmitted through the second reflector, take 
place, the former in the plane of second reflexion and 
the latter in a plane at right angles to it; and that important 
the effect of the second reflector is, therefore, to twist,. 
as it were, the planes of polarization of these component 
w r aves in opposite directions, that of the reflected wave 
through an angle which measures the rotation of the 
second reflector about the axis of the tubes, and that of the 
refracted wave through an angle which is its complement. 

It thus appears that a beam of homogeneous light re¬ 
flected from, or refracted through, a plate of glass, be¬ 
ing incident under an angle equal to 5 6°, immediately 





330 


NATURAL PHILOSOPHY. 


Characteristics 
of plane 
polarized light; 


Effects when the 
angle of 
Incidence differs 
from that of 
polarization. 


Apparatus. 


Position of the 
plane of 
polarization 
determined; 


Analyzer 
revolved; 


Intensity of 
rellected beam; 


acquires opposite properties, with respect to reflexion and 
refraction, on sides distant from each other equal to 90°, 
measuring around the beam ; and the same properties at 
distances of 180° ; and these among other properties dis¬ 
tinguish plane polarized light. 

We have supposed the angle of incidence 56°,. if it 
were less or greater than this, similar effects would be ob¬ 
served, though less in degree; or, in other words, the 
waves first deviated would be elliptically polarized, the 
eccentricity of the elliptical orbit increasing as the angle 
approaches more and more to that of polarization. 


Fig. 93. 



The plate MN is called the polarizer, and M r 'N\ the 
analyzer. The position of the plane of polarization in 
any plane polarized wave, is readily ascertained by the 
total reflexion which takes place from the analyzer, when, 
the polarized beam being incident under the polarizing 
angle, the plane of the analyzer is perpendicular to it. 
Starting from this position of the analyzer, with respect 
to the plane of polarization, and calling a, the angle be¬ 
tween the plane of polarization and that of second inci¬ 
dence, which is equal to the angle through which the 
analyzer has at any time been turned about the first 
reflected or polarized beam ; A, the intensity of this beam, 
and /, the variable intensity of that reflected from the 
analyzer in its various positions, the formula 

1= A cos 2 a,.(109) 






ELEMENTS OF OPTICS. 


831 


will express, for uncrystallized media, the law according La w expressed 
to which a polarized beam will be reflected from the byformula; 
analyzer when the angle of incidence is equal to that 
of polarization. 

According to this law, if we conceive a wave of com- ™ s law applied 
mon light as it emanates from any self-luminous body, ^“ mon 
to be compounded of two w r aves polarized in planes at 
right angles to each other, that is, supposing the orbital 
motion of the molecules to arise, as they will, from two 
component rectilinear motions at right angles to each 
other, Equation (107), we should have for the intensity 
of reflexion from a reflector, 

I 4“ I' — A . COS 2 a + A. . COS 2 (90°-a ) = A, Consequence; 

in which I and denote the intensity of reflexion of 
the two component polarized w r aves; whence, the inten¬ 
sity of the reflected wave will be the same on whatever conclusion; 
side of the incident beam the analyzer be presented. 

§135. What has been said of the effects of glass Polarizing anglo 
on light is equally true of other transparent homogene- varies with the 
ous media, except that the polarizing angle, which is con¬ 
stant for the same substance, differs for different bodies. 

It is found, from very numerous observations, that the 
tangent of the maximum polarizing angle is always equal 
to the refractive index of the reflecting medium taken in 
reference to that in which the wave is reflected; thus, 
calling the relative index m , and the polarizing angle 9 , 
we shall have, 


tan 9 = m .(110) Samelnformof 

an equation; 

Example. Let it be required to find the polarizing an¬ 
gle when light is moving in water and reflected from 
glass. The refractive indices for water and glass are Example; 
1,336 and 1,525, respectively, hence, 




332 


NATURAL PHILOSOPHY. 


Numerical data; 


Result. 


Supposition; 

/ 


Consequence. 


Table of 
polarizing angles. 


No perfect 
polarization for 
white light; 


And a tint will 
be reflected from 
the analyzer; 


"What is 
understood by 
polarizing angle 
for white light; 


1,525 
m — -2— 
1,336 


1,1415 = tan 9 , 


or 

9 = 48° 47'. 

If the refractive indices of the media were equal, we 
should have 


711 = 1 


9 = 45°. 

The following are the values of 9 , for the different sub¬ 
stances named, the wave being reflected in air. 


Water,. 

0 

CO 

11 ' 

Crown glass, - - - 

56° 

55' 

Plate glass, - - - - 

57° 

45' 

Oil of Cassia,- - - 

58° 

Oi 

CO 

Diamond, - - - - 

68 ° 

6 ' 


§ 136. It is obvious that according to the law expressed 
by Equation (110), there can be no such thing as per¬ 
fect polarization by reflexion in white light, since the re¬ 
fractive index is not the same for the different colors; 
and hence there can never be total absence of light at 
the analyzer; but a certain tint will be reflected , whose 
intensity will depend upon the dispersive power of the 
medium. For bodies of very high refractive powers, 
which are also, in general, highly dispersive, we must, 
therefore, understand by the polarizing angle for white 
light, that angle of incidence at which the reflected light 
approaches nearest to perfect polarization. This angle 
being ascertained for opaque bodies by experiment, the 





ELEMENTS OF OPTICS. 


333 


relation expressed by Equation (110), furnishes the means Method of 
of ascertaining their refractive indices. Thus, the maxi- refracts indices 
mum polarizing angle for steel is a little over 71°, the of °P a( i uebodies - 
natural tangent of which is 2,85, which is, therefore, ac¬ 
cording to the law, its refractive index; the polarizing 
jr angle for mercury is about 76° 30', and its refractive 
index, consequently, 4,16. 


§ 137. We have spoken, thus 
far, only of the action at the first 
surface of the glass plate; it is 
found that the light reflected at 
the second surface is as perfectly 
polarized as that reflected at the 
first, and in the same plane, when 
the faces of the plate* are parallel. 
This is a consequence of the same 
law for, 


Fig. 97. 



Light reflected 
at the second 
surface also 
polarized; 


m = tan 9 = 


sin 9 
cos 9 


sin 9 
sin 9 ' 


Equation; 


hence, 


cos 9 = sin 9 ' Eelation; 

or 9 ' is the complement of 9 , and the first reflected beam 
is perpendicular to the first refracted. 

Moreover, 

JL __ = cot 9 = tan 9 ' 

m tan 9 


but — is the index of the wave passing out of the glass; 
m 

hence <p' is the maximum polarizing angle for the second 

r A Polarizing angle 

surface. for second 

If a series of parallel plates be employed in the form surface tod- 









334 


NATURAL PIIILOSOniY. 


Effect of a pile of of a pile, the light reflected from the second surfaces 
plates. coming off polarized in the same plane, a polarized beam 

of great intensity may be obtained. This intensity can, 
however, never exceed half that of the incident beam, 
no matter how great the number of plates employed. 


Effect of 
repeating the 
reflexions at 
angles different 
from that of 
polarization. 


§ 138. Although a wave of homogeneous light is but 
elliptically polarized when reflected once at an angle dif¬ 
fering from that of polarization, yet by repeating the re¬ 
flexions a sufficient number of times, the ellipse may be 
reduced to a right line, in which case the light will be 
plane polarized; and in doing this, it is not necessary 
that the reflexions take place at the same angle of inci¬ 
dence, but some may be above and some below the po¬ 
larizing angle. In general, the number of reflexions will 
increase as the angle of incidence recedes from that of 
polarization on either side. 

The same remarks will apply to light polarized by re¬ 
fraction. 


POLARIZATION BY ABSORPTION. 


Polarization by 
absorption; 


Experiment with 
a plate of 
tourmaline; 


Fig. 98. 


§ 139. A plate of Tourmaline , about of an inch thick, 

cut parallel to the axis, possesses the property of inter¬ 
cepting that component of common light whose vibrations 
take place in a plane parallel to the axis, and of transmit¬ 
ting the other. This latter will, of course, be polarized in a 
plane at right angles to the axis 
of the crystal. If, therefore, light 
previously plane polarized, be in¬ 
cident upon the plate with its 
plane of polarization perpendicu- 
lar to the axis, it will be wholly \0~Q~5~601 
transmitted; but if parallel, it 
will be wholly absorbed or inter¬ 
cepted. This is another property 


ffiWl/W 





ELEMENTS OF OPTICS. 


335 


by which plane polarized light may be distinguished, a characteristic 
Hence, two plates of tourmaline form a most convenient ]ight . 
apparatus for experimenting with polarized light when so 
arranged as to be capable of turning about a common 
axis, the one being used to polarize light, the other to 
analyze it. Plates of agate and some varieties of quartz A PP aratus of 
possess similar properties. plateSt 


DOUBLE REFRACTION. 


§ 140. In treating of the transmission of light through Bodies in which 
different media, we have regarded the ether of the latter 
as possessing the same density and the same elasticity in all spherical : 
directions; in which case the luminous waves proceeding 
from any point, will always be spherical. But there is a 
large class of bodies in which neither of the above condi¬ 
tions exists. This class embraces all crystallized media ex¬ 
cept those whose primitive form is the cube , the octohedron , 
and the rhomboidal dodecahedron / also all animal sub- Those in which 
stances among whose particles there is a tendency to reg-^ 3 ™^. 11041 * 5 
ular arrangement; and, in general, all solids in a state 
of unequal compression or dilatation. 

As has been stated already, (§ 42, Acoustics,) the most 
general hypothesis, consistent with permanence of figure* 
that may be made with regard to the internal constitution 
of such bodies, is that which attributes a difference of elastic 
force in three directions at right angles to each other. The 
law of the elastic force, in directions inclined to these, is 
given by the equation of the surface of elasticity, and the constitution of 
shape of a wave propagated through such a body, is defined ^ e a ” ine 
by Equation (16) of the same article. It must not be in¬ 
ferred, however, that the lines represented by a, b and c, 
in that equation, and denominated axes of elasticity, have 
any particular location. They may have their origin any¬ 
where within the body, but must always be drawn in the 
same direction through it. Indeed, the principles of crys- 



336 


NATURAL PHILOSOPHY. 


Constitution of 

crystalline 

bodies. 


Principal sec¬ 
tions of wave 
surface. 


Model of wave 
surface. 


tallization lead us to admit that the arrangement of the 
molecules of a crystalline body, is similar in all parallel 
lines throughout the crystal, and the same property must 
belong to the ether within it, if, as we have every reason 
to presume, its elasticity be dependent upon that of the 
crystal. 


§ 141. The figure of the wave surface, given by Equa¬ 
tion (16), is studied to best advantage by taking its sections 
by the planes of the axes of elasticity. Supposing a > b, 
and b > c, these sections, by the planes b c, a c, and a b : are 
respectively, 


x — 0 ; (y 2 + z 2 — a 2 ) (6 2 y 2 + c 2 z 2 — 6 2 c 2 ) = 0, 

y — 0 ; (z 2 + x* — 6 2 ) (c 2 z 2 -J- a 2 x* — c 2 a 2 ) = 0, 

z = 0; (x 1 -f- y 2 — c 2 ) (a 2 x 2 + y 2 — a 2 6 2 ) = 0, 

The first gives a circle and an ellipse, the latter lying 
wholly within the former; the third gives the same kind 
of curves, but the ellipse wholly enveloping the circle; 
the second gives the same kind of curves, intersecting one 
another in four points. This last is the most important. 
It is the section parallel to the axes of greatest and least 
elasticities. 

It thus appears that the general wave surface, defined by 
Equation (16), “ Acoustics,” con¬ 
sists of two nappes , the one wholly 
within the other, except at four 
points, where they unite, and at 
each of which they form a double 
umbilic, somewhat after the man¬ 
ner of the opposite nappes of a 
very obtuse cone. The figure 
represents a model of the wave 
surface, cut by the planes of the axes. The sections show 
the umbilic points, as well as the general course of the 









ELEMENTS OF OPTICS. 


387 


nappes, by the removal of a pair of the resulting diedral 
quadrantal fragments. 


§ 142. Taking the section by the plane a c, the semi- 
transverse axis of the ellipse will be equal to a , and the 
radius of the circle to b. Joining, by diagonal lines, the 
points of intersection of the ellipse and circle, and denoting 
the cosines of the angles which these lines make with the 
axis a, by a y and a„, with the axis b by and , and 
with the axis c by y t and y tn it is shown, in the “Analyt¬ 
ical Mechanics,” § 320, that 


Directions of 
equal wave 
velocities. 



And denoting by u / and u u the angles which any arbitrary 
line, drawn from the origin, makes with these diagonal 
lines, then will the velocities, denoted by V r and V r ^ of 
the points of the two nappes on this line, be given, (Ana¬ 
lytical Mechanics, § 320,) by 


jF? = h (l + j.) + i (l ~ l) ■ ( cos «- • 003 + sin • sin “")> 

Reciprocal of 

^ ’ wave velocities, 

= i + j.) + i - tf) • ( cos • cos - sin M ' • sin u ‘h 

and by subtraction, 

1 

V ~» 

Y r 2 

Now, 


are the retardations of wave velocity. As long as a and c 
differ, the second member can only reduce to zero, when 
22 


1 /I 1 . . 

V? = [?-a’)- SmU '- SmU " 


fllll Spherical 
7 lemniscates. 


y- and y~ 

r r. r r.i 







338 


NATURAL PHILOSOPHY. 


Biaxal bodies. 


Equal elasticity 
on two of the 
axes. 


Locus of equal 
wave retardation 
circular. 


Ellipsoidal and 

spheroidal 

waves. 


u / or u n is zero; whence it appears that, as a general rule, 
every direction except two is distinguished by transmit¬ 
ting two waves, one in advance of the other. The two 
directions which form the exceptions are in the plane ol 
the axes of greatest and least elasticity, and make with 
these axes the angles of which the cosines are a, and y / , 
a y/ and y u . In these directions the waves will travel with 
equal velocities. 

Any direction along which the component waves travel 
with equal velocities is called an axis of equal wave velocity. 
All bodies in which the elasticities in three rectangular 
directions differ, possess, therefore, two of these axes, and 
are called biaxal bodies. The retardation of one compo¬ 
nent wave over that of the other, will vary with the incli¬ 
nation of the direction of its motion to the axis of equal 
wave velocity; and Equation (111) shows that the loci of 
equal retardations will be arranged in the form of spherical 
lemniscates about points on the axes as poles. 

§ 143. If b = c, then will 

“/=!; 7 /= 0 ; 

the axes will coincide with one another and with the axis 
a, that is, with x\ u J will equal u jn and, Equation (111), 



Also, Equation (16) of the general wave surface becomes 

(x 2 + y 2 + 2 2 — c 2 ) [a 2 x 2 + c 2 (y 2 + z 2 ) — a 2 c 2 ] = 0; 

and the wave surface will be resolved into the surface of a 
sphere, and that of an ellipsoid of revolution. Making 
u, = 0, it will be seen, from Equation (112), that these 
waves travel with equal velocities in the direction of the 
axis a. For any other value for u, since u / = u n , we have 
cos u t cos u {i + sin u ( sin u u = 1; whence 



ELEMENTS OF OPTICS. 


339 



and it hence appears, that the velocity of one of the com¬ 
ponent waves will be constant throughout its entire extent, 
while that of the other will be variable from one point to 0rdinaryarul 

r extraordinary 

another more and more remote from the axis. The first is waves, 
called the ordinary , the second the extraordinary leave. 

If c be greater than a, then will the ellipsoid be prolate; 
if less than a, it will be oblate. There is but one direction 
which will make Vf = V r 2 2 , and that is coincident with the 
axis a. Bodies in which this is true have but one axis of 
equal wave velocity, and are called uniaxal bodies. 

From Equation (112) it appears, that the loci of equal 
retardations are concentric circles, of which the common 
centre is on the axis of equal wave velocity. 

§ 144. The phenomenon which certain bodies thus pre¬ 
sent, of resolving the living force impressed upon its 
ethereal molecules into two components, and of transmit- tion> 
ting these components with different velocities, is called 
double refraction. 

The index of refraction, is the ratio which the velocity of 
a wave in the medium of incidence bears to that in the 
medium of intromittance; and this ratio is the same as the 
sine of the angle of incidence to that of refraction. It 
therefore follows, from this discussion, that a wave of com¬ 
mon light, falling upon the surface of biaxal or uniaxal 
bodies, will divide into two parts, and the parts will take 
different directions through the body; and, hence, all ob- Bodies seen 
jects seen through such bodies will appear double. 

The components which come from the resolution of a 
common wave are polarized. 

Grlauberite, nitrate of potassa, arragonite, sulphate of Instancesof 
baryta, mica, sulphate of lime, topaz, carbonate of potassa, biaxal bodies, 
and sulphate of iron, are among the biaxal bodies. Ice¬ 
land spar, carbonate of zinc, phosphate of lead, tourmaline, Instances of 
quartz, emerald, beryl, and ruby, are some of the uniaxal uniaxal bodies. 



NATURAL PHILOSOPHY. 


340 


Obl ire mid 
Prolate waves. 


Plane of princi¬ 
pal section. 


Double 
refraction in 
a particular 
instance 
considered; 


Two plane 
polarized waves 
within the 
crystal; 


If the second 
surface be 
parallel to the 
first, no double 
refraction 
observed; 


class. In Iceland spar, c is less than a , and in quartz, (six- 
sided prisms,) c is greater than a. In the first case the 
extraordinary wave is oblate , and in the second prolate. All 
these bodies are distinguished from one another by greater 
or less peculiarities of crystalline form; but the second 
class differs from the first by the exhibition of some one 
remarkable line of symmetry, showing a great difference 
in the law of internal molecular arrangement between the 
classes. 

§ 145. A plane through either two of the axes of elas¬ 
ticity is called a plane of principal section. In a biaxal body, 
there are but three of such planes, but in uniaxal bodies 
there are an infinite number; for any plane containing the 
axis of equal wave velocity, which is one of the axes of 
elasticity, will also contain another of these axes, all lines 
at right angles to that of equal wave velocity being lines 
of equal elasticity. 

§ 146. To illustrate how double refraction takes place 
in a particular instance, take, for example, the simple case 
of a beam of light proceeding from an indefinitely distant 
point, and falling perpendicularly on the surface of an uni¬ 
axal crystal, cut parallel to the axis. The incident wave 
being plane, and parallel to the surface of the crystal, the 
vibrations are also parallel to the same surface, and will 
be resolved into two component vibrations, the one par¬ 
allel and the other perpendicular to the axis of the crystal. 
Now, the elasticity brought into play by these two sets of 
vibrations being different, they will be propagated with 
different velocities; and there will be two waves within 
the crystal polarized in planes at right angles to each 
other. If the second face of the crystal be parallel to the 
first, the two waves will emerge parallel, resuming the 
velocity which they had before incidence; they will, there¬ 
fore, be unequally accelerated, but will retain their paral¬ 
lelism after emergence, the only effect being to cause one 
to lag behind the other. But when the second face of the 



ELEMENTS OF OPTICS. 


341 


crystal is oblique to the first, it will also be oblique to the if the second 
wave fronts, and this obliquity will make their unequal oblique to tto 
change of velocity apparent by causing the waves to take first, double 
different directions; there will, in this case, be double^ f ™^ ionWl11 
refraction at emergence. 

One of the component waves in Iceland spar is propa- Form of the 
gated equally in all directions, and is, therefore, spherical com ponent 

• _ ... . . waves in Iceland 

m lorm when proceeding from a point m the crystal; the 8 pa r. 
other is propagated unequally in different directions, the 
form of the wave being that of an oblate spheroid of revo¬ 
lution, whose shorter axis coincides with the optical axis 
of the crystal. 

Now, the radius of the ellipsoidal wave is always Eelationt)et ' vcer * 

. the radii of the 

greater than that of the spherical wave, except when ordinary ftml 
the refracted ray coincides with the axis; and these radii extraordinary 
being described in the same time, may be taken as the 
measures of the velocities of wave propagation in the 
extraordinary and ordinary waves. The refractive in¬ 
dex being equal to the ratio of 
the velocity before incidence, to 
that within the crystal, the extra¬ 
ordinary index will be variable, 
and less than the ordinary index. 

But the index of refraction being 
also equal to the ratio of the sine 

of the angle of incidence to that of refraction, the extra¬ 
ordinary ray must always be thrown farther from the axis ^ ^ 
than the ordinary ray; and the extraordinary index of extraordinary 
refraction will have its minimum value when the extra - index w,n be a 

minimum. 

ordinary ray is perpendicular to the axis. 


Fig. 100 . 



Extraordinary 
index variable 


§ 147. With rock crystal , which oc- Fig. 101 . 
curs in the form of hexagonal prisms, 
terminated with six-sided pyramids, the 
case is just reversed; the ellipsoidal 
wave is prolate, its longer axis coin¬ 
ciding with the optical axis of the 
prism, and being equal in length to the radius of the 



Eock crystal; 


Properties the 
reverse of those 
of Ireland spar. 







342 


NATURAL PHILOSOPHY. 


Doubly refract¬ 
ing substances 
classified. 


Positive 

crystals. 


Negative 

crystals. 


Biaxal crystals, 
with inclination 
of their axes. 


spherical wave; the extraordinary ray is always found 
between the ordinary ray and the axis, as if drawn towards 
the latter; and the extraordinary index is a maximum 
when the extraordinary ray is perpendicular to the axis. 
These circumstances have given rise to a division of doubly 
refracting substances into two classes,distinguished by their 
axes, which are said to be positive when the extraordinary 
ray is between the ordinary ray and the axis, as in the 
case of rock crystal; and negative when the positions of 
these rays are reversed with respect to the axis, as in Ice¬ 
land spar. 


TABLE OF A FEW POSITIVE CRYSTALS. 


Zircon. 

Quartz. 

Tungstate of zinc. 
Stannite. 

Boracite. 


Hydrate of magnesia. 
Ice. 

Hydrosulphate of lime. 
Dioptase. 

Sulphate of potassa. 


TABLE OF SOME NEGATIVE CRYSTALS. 


Iceland spar. 

Carbonate of lime and magnesia. 
Carbonate of lime and iron. 
Tourmaline. 

Ruballite. 

Sapphire. 

Ruby. 

Emerald. 


Beryl. 

Apatite. 

Mica. 

Phosphate of lead. 
Arseniate of copper. 
Cinnabar. 

Phosphate of lime. 
Idocrase. 


TABLE OF A FEW BIAXAL CRYSTALS, WITH THE INCLINATION OF 
THEIR AXES. 


Sulphate of nickel . . 3 00 

Talc.7 24 

Hydrate of barytes . 13 18 

Arragonite . . . . 18 18 

Borax. 28 42 

Sulphate of magnesia. 37 24 
Sulphate of barytes . 37 42 
Spermaceti . . . . 37 40 


Stilbite.41 42 

Sulphate of nickel. . 42 04 

Topaz. 50 00 

Sulphate of lime . . 60 00 
Feldspar ..... 63 00 
Carbonate of potassa . 80 30 

Cyanite.81 48 

Sulphate of iron . . 90 00 





ELEMENTS OF OPTICS. 


843 


§ 148. If a plane wave 
W IF', of common light be 
incident on the upper sur¬ 
face of a crystal of Iceland 
spar to which it is parallel, 
this wave will be resolved 
into two components, one of 
which will take the direction 
of and be normal to an ob¬ 
lique line P e, and will be 
refracted according to the 
extraordinary law; the other 
will preserve its original 
course and pass through without deviation. These waves 
will both leave the crystal normal to that plane of prin¬ 
cipal section which is perpendicular to its upper face, 
the waves themselves becoming parallel; each will be 
plane polarized, the plane of polarization of the ordinary 
wave coinciding with the plane of principal section just Effect of this 
named, and that of the extraordinary wave being at crystaL 
right angles to it. 

If these component waves be received upon the upper Emergent waves 
surface of a second crystal of the same kind, and whose ^Jdcrystei of 
optical axis is parallel to that of the first, they will take Iceland spar : 
the directions e' e" and o' o'\ parallel, respectively, to the 
directions P e , and P o , and will not be again divided, 
the first undergoing extraordinary, and the latter ordi¬ 
nary refraction; and if the crystals be of equal thick¬ 
ness, the distance d' 0 ", will be double e 0 . If either or 
both of the component waves whose directions are e e\ 
and 0 o\ had been polarized by reflexion, refraction or 
absorption, the action of the second prism would have 
been the same; this is, therefore, another characteristic 
property of plane polarized light, viz.: that it will not un¬ 
dergo double refraction when its plane of polarization is 
either parallel or perpendicular to the plane of principal sec- Another 
tion ; being in the former case wholly refracted according to 
the ordinary , and the latter according to the extraordi- light 


Fig. 103. 
jS 



Plane wave of 
common light 
incident upon a 
crystal of Iceland 
spar; 










NATURAL PHILOSOPHY. 


344 


Reverse true for nary law. The reverse would have been the case if the 
positive crystals. cr y S tal, like quartz, had possessed a positive axis. 


§ 149. "When the crystal 
If W', is turned around on 
its base so that the prin- 
The second c ipal sections of the crys- 

crystal supposed 

to turn on its tals, which are normal to 
base; .the upper surfaces, make 

an angle with each other, 
each of the component 
waves of which the direc¬ 
tions are oo r and ee\ will 
be again divided into an 


ordinary and extraordi¬ 
nary wave, whose relative 
intensities will depend up¬ 
on the inclination of the 
principal sections to each 
other. To avoid complica¬ 
tion, let us suppose the 

Effect on the wave moving along P e, to 

ordinary wave; . , , . 

be arrested by sticking a 
piece of wafer to the lower 
surface of the first crystal 
at e ; then will the intensi¬ 
ties of the portions into 
which the wave moving 
along o o\ is divided by 
the second crystal, be ex¬ 
pressed by the formulas 


Effect on the 

ordinary 

wave; 


0 0 ~ A . COS 2 a 
O e = A. sin 2 a 


.(113) 


Fig. 103. 
>S 



Fig. 104. 


O 



Fig. 105. 



Wherein A represents the 
intensity of the wave o o '; 




















ELEMENTS OF OPTICS. 


345 


a, tlie angle made by the principal sections of the crys- Notation; 
tals; 0 0 , the intensity of the ordinarily refracted wave; 
and <9 C , that of the wave refracted according to the ex¬ 
traordinary law. 

Removing the wafer from 0 , and calling E e and E 0 
the intensities of the extraordinary and ordinary waves 
into which the wave moving on P e is separated by the 
second crystal, and B its intensity on leaving the first 
crystal, we shall, in like manner, have 


E c = B . COS 2 a 
E 0 = B . sin 2 a 


• (H4) 


Components of 
the extraordinary 
wave; 


Taking the sum of the four emergent waves, there will 
result, 


0 0 + O e + E e + E 0 — A 4- B. 


Sum of the four 
emergent waves. 


The waves 0 0 and 6> e , in Equations (113), are always 
found to be polarized, the former in the plane of princi¬ 
pal section of the second crystal, the latter in a plane at 
right angles to it; and the same Remark being applica¬ 
ble to Eo and E e) in Equations (114), it follows that the 
planes of polarization of 0 Q and E 0 will be parallel to ^1°™ 
each other, as also those of 0} and E e . polarization. 


CIRCULAR POLARIZATION. 


§ 150. All questions of polarization are directly con¬ 
cerned with the shape of the molecular orbits and thef°^ 10n 
directions of the molecular motions in these orbits. It is molecular orbits, 
shown (Analytical Mechanics, §§ 340-343): 



346 


NATURAL PHILOSOPHY. 


Circular orbits 
produced. 


Directions of 

molecular 

motion. 


Waves oppositely 
polarized, and 
plane of 
crossing. 


Composition of 
waves. 


Resolution of 
waves. 


1st. That two waves, plane polarized, will, by their simul¬ 
taneous action upon the same molecule, cause it to move 
uniformly in a circular path, provided they be of the same 
length and intensity, and the same phases m each are sep¬ 
arated in the direction of wave motion by one quarter, or 
any odd multiple of a quarter, of wave length. 

2d. That the molecular motion in this orbit will take 
place from right to left or left to right, as viewed from the 
same point, depending upon the directions of its motions 
in the component waves at the instant of their simultane¬ 
ous action. 

3d. Two circularly polarized waves, in which the mo¬ 
lecular motions are in opposite directions, are said to be 
oppositely polarized; and supposing the orbits in two such 
waves to coincide, a plane perpendicular to the w&ve front, 
through their common centre and the place of the mole¬ 
cule at the instant these waves begin their simultaneous 
action upon it, is called the plane of crossing. 

4th. That the simultaneous action of two oppositely po¬ 
larized waves, give a resultant wave polarized in the plane 
of crossing, and of which the intensity is double that of 
either component. 

5th. Conversely, a plane polarized wave may.be re¬ 
solved into two equal and oppositely polarized waves. 

The resolution and Composition of plane and circularly 
polarized waves, are well illustrated by two and four in¬ 
ternal reflexions from the faces of Fresnel’s rhomb of St. 
Gobain’s glass. 


Effect of metallic §151. We might naturally conjecture that the effects 

reflectors on _ 

plane polarized produced by metals upon the reflected light 'would be 
lighL analogous to the phenomena of total reflexion by glass 

and other transparent substances,—there being no tran¬ 
smitted wave in either case. It is accordingly found 
that when a plane polarized wave is incident upon a 
metallic reflector, the reflected light is elliptically 
polarized ; the laws of the phenomena are, however, dif¬ 
ferent from those of total reflexion from transparent media. 



ELEMENTS OF OPTICS. 


847 


§ 152. There are many substances whose molecular 
structure is such as to resolve a plane polarized wave into Substances tliat 
two component waves circularly and oppositely polarized, of polarization, 
and transmit them with different velocity. In consequence, 
the phases peculiar to each at the instant of resolution will 
separate in the direction of wave propagation, and at 
emergence from the substance, will have their plane of 
crossing inclined to its first position, which was coincident 
with the primitive plane of polarization,—the final effect 
being, to give the plane of polarization of the resultant 
emergent wave an inclination to its position before enter¬ 
ing, as though this plane had been revolved about a line 
normal to the wave front. 

It is shown, (Analytical Mechanics, §§ 842, 843,) that 
the law of this rotation is given by the equation 

V,.t_V .t 

2* x ’ 

in which V / is the angular velocity, V the velocity of wave 
propagation, X the length of the wave, t the time of the 
wave’s motion in the body, and « the ratio of the circum¬ 
ference of a circle to the diameter. The first member is 
the arc, expressed in circumferences, described by the 
molecule while the wave is moving through a thickness 
V. t of the medium. So that a wave, compounded of 
many components having different wave lengths, but all 
polarized on entering a medium, may emerge with the 
planes of polarization of its several components so twisted 
through different angles as to diverge from a common line 
perpendicular to the wave front. Crystalline and vegeta¬ 
ble bodies furnish many examples of this. A piece of 
quartz, of a peculiar kind, is known to twist the plane of 
the extreme red wave through an angle of 17° 29' 47 A/ , Effects of some 
and of the extreme violet, 44° 04' 58 // , for each 0.04 of an vaneties of 
inch. Different varieties of the plagiedral quartz turn the 
plane of polarization in opposite directions, and a connec¬ 
tion exists between this property and the right or left 


Law of tb© 
rotation. 



348 


NATURAL PHILOSOPHY. 


Other bodies 
possess the same 
property; 


Formula for a 
combination 


Notation 
explained; 


1 


Applies also te 
liquids. 


Introductory 
remarks; 


First discoveries. 


handed direction in which certain small faces lean around 
the summit of the crystals; and if two of these bodies be 
interposed, the arc of rotation is that due to the sum or dif¬ 
ference of their thicknesses, according as they exert their ac¬ 
tion in the same or opposite directions. Or, more generally, 

R T = r t 4- r* t’ + r" t" + &c., 

in which R is the rotation due to the combination; T 
its entire thickness ; r , r\ &c., and t, t\ &c., the corres¬ 
ponding quantities answering to the several individuals 
of the combination; the products entering the expression 
with the same or different signs, according as the diffe¬ 
rent media tend to turn the plane of polarization in the 
same or different directions. This formula is found to 
hold good not only with solid crystals, but also with 
liquids possessing this property, when mixed together. 


CHROMATICS OF POLARIZED LIGHT. 

§ 153. Having explained the general phenomena of 
polarization and double refraction, we pass to the consid¬ 
eration of the effects produced when polarized light is 
transmitted through crystalline substances. The phe¬ 
nomena displayed in such cases, are among the most 
splendid in optics; and when we consider that through 
these phenomena we are enabled almost to view the in¬ 
terior structure and molecular arrangement of natural bo¬ 
dies, the importance of the subject will be apparent. 

The first discoveries in this department of science were 
made by Arago, in the year 1811, and the subject has 
since been successfully prosecuted by some of the first 
philosophers of Europe. 

§ 154. We have seen that when a wave of light, po¬ 
larized by reflexion, is incident upon the analyzer under 
the polarizing angle, no reflexion will take place when 




ELEMENTS OF OPTICS. 


349 


the plane of incidence on the analyzer is perpendicular Effect of 
to that on the polarizer. Now, if between the two re- transraittin s 
hectors we interpose a plate of any double-refracting sub- throughl 18 ^ 
stance, the power of reflexion at the analyzer is suddenly ^bie-refracting 
restored, and a portion of the light is reflected, the quantity Cry8tal 
^depending on the position of the interposed crystal; and 
by this property the double-refracting structure has been 
detected in a vast variety of substances, in which the sep¬ 
aration of the two waves was too small to be directly per¬ 
ceived. 


§ 155. In order to analyze this phenomenon, let the crys- These effects 
talline plate be placed so as to receive the polarized wave analyzed h r 
parallel to its surface, and let it be turned round in its crystal in its own 
own plane. We shall then observe that there are two plane; 
positions of the plate in which the light totally disap¬ 
pears, and the reflected wave vanishes, just as if no 
crystal had been interposed. These two positions are 
at right angles to one another; and they are those in 
which the principal section of the crystal coincides with When no 
the plane of first reflexion , or is perpendicular to it. reflected from the 
When the plate is turned round, from either of these analyzer; 
positions, the light gradually increases, until the princi- a 

pal section is inclined at an angle of 45° to the plane maximum, 
of first reflexion, when it becomes a maximum. 


§ 156. In these experiments the reflected light is white. Colors produced 
But if the interposed crystalline plate be very thin, pf at v e e 3 rythm 
most gorgeous colors appear, which vary with every 
change of inclination of the plate to the polarized wave. 

Mica and sulphate of lime are very appropriate for the 
exhibition of these beautiful phenomena, because they su]pliateoflime . 
can be readily divided by cleavage into laminse of almost 
any required thinness. If a thin plate of either of these 
substances be placed so as to receive the polarized wave 
parallel to its surface, and be then turned round in its Appearances 
own plane, the tint does not change, but varies only in 
intensity; the color vanishing altogether when the prin-substances ; 



350 


NATURAL PHILOSOPHY. 


When the light 
disappears and 
when it is a 
maximum. 


cipal section of the crystal coincides with the plane of 
primitive polarization, or is perpendicular to it,—and, 
reaching a maximum , when it is inclined to the plane of 
primitive polarization at an angle of 45°. 


The crystal fixed g 157. If, on the other hand, the crystal be fixed, and 
turned 6 ; ana,yZer the analyzer be turned, so as to vary the inclination of 
the plane of the second reflexion to that of the first, 
the color will be observed to pass, through every grade 
of the same tint, into the complementary color; it being 
positions giving always found that the light reflected in any one position 
complementary 0 f the analyzer is complementary , both in color and in¬ 
tensity, to that which it reflects in a position 90° from 
the former. This curious relation will appear more evi¬ 
dently, if we substitute a double refracting prism for 
the analyzer; for the two waves refracted by the prism 
have their planes of polarization—one coinciding with the 
Doubierefracting principal section of* the prism, and the other at right an- 
C T?! * , gles to it, and are therefore in the same condition as the 

the analyzer; light reflected by the analyzer, with its plane of reflex¬ 

ion successively in these two positions. In this manner 
the complementary colors are seen together, and may be 
easily compared. But the accuracy of the relation sta¬ 
ted is completely established by making these two waves 
partially overlap ; for, whatever be their separate tints, it 
causing the will be found that the part in which they are superposed 
tints to overlap, is absolutely white . 

Effect of plates of g 753 . ^yhen laminae of different thicknesses areinter- 

variable 

thicknesses ; posed between the polarizer and analyzer, so as to re¬ 
ceive the polarized wave parallel to their surfaces, the tints 
are found to vary with the thickness. The colors pro¬ 
duced by plates of the same crystal, of different thick¬ 
nesses, follow, in fact, the same law as the colors reflect¬ 
ed from thin plates of air; the tints rising in the scale 
Law followed by as the thickness is diminished, until finally, when this thick¬ 
ness is reduced below a certain limit, the colors disap¬ 
pear altogether, and the central space appears lilack , as 






ELEMENTS OF OPTICS. 


351 


when the crystal is removed. The thickness producing Results of 
corresponding tints is, however, much greater in crystal- experiments; 
line plates exposed to polarized light, than in thin plates 
of air, or any other medium of homogeneous structure. 

The black of the first order appears in a plate of sul¬ 
phate of lime, when the thickness is the 2 cV o °f an inch 5 
between 2 oV 0 an d sV °f an inch, we have the whole 
succession of colors of Newton’s scale; and when the 
thickness exceeds the latter limit, the transmitted light 
is always white . The tint produced by a plate of mica, different 
in polarized light, is the same as that reflected from a substances 

i ° 1 , compared. 

plate ot air ot only the T £ 7 th part ot the thickness. 

The same subject has been investigated for oblique in- 0blique 
cidences, and the law r s which connect the tint developed incidences, 
wflth the number of wave lengths and parts of a length 
within the crystal, for a wave of given refrangibility, 
have been determined, both for uniaxal and biaxal crys¬ 
tals. 


§ 159. Let us now apply the principles already estab- Ap pi ication 4 
lished, to explain the appearances. preceding 

It has been shown, that a wave of common light, on pnnciples ’ 
entering a crystalline plate, is resolved into two waves, 
which traverse the crystal with different velocities, and in 
different directions. One of these waves, therefore, will 
lag behind the other, and they will be in different phases 
of vibration at emergence. When the plate is thin, this Preliminary 
retardation of one wave upon the other will amount only remark8; 
to a few wave lengths and parts of a length; and it 
would, therefore, appear that we have here all the condi¬ 
tions necessary for their interference , and the consequent 
production of color. 

But here we are met by a difficulty. So far as this An apparent 
explanation goes, the phenomena of interference and 0 f dlfficu,t y ans « s • 
color should be produced by the crystalline plate alone, 
and in common light, without either polarizing or ana¬ 
lyzing plate. Such, however, is not the fact; and the 
real difficulty in this case is,—not so much to explain 



352 


NATURAL PHILOSOPHY. 


Its solution; 


enquiry 

suggested. 

Experimental 
researches on th 
interference of 
polarized light; 


Rules deduced. 


Experimental 

illustration. 


how the phenomena are produced, as to show why they 
are not always produced. 

In seeking for a solution of this difficulty, it may be 
remarked, that the two waves, whose interference is sup¬ 
posed to produce the observed results, are not precisely 
in the condition of those whose interference we have 
hitherto examined; they are polarized , and in planes at 
right angles to each other. We are led, then, to inquire 
whether there is anything peculiar to the interference 
of polarized waves which may influence these results; 
and the answer to this inquiry will be found to remove 
the difficulty. 

§ 160. The subject of the interference of polarized light 
was examined, with reference to this question, by Fres¬ 
nel and Aeago, and its laws experimentally developed. 
It was found that two waves of light, polarized in the 
same plane, interfere and produce fringes, under the same 
circumstances as two waves of common light;—that 
when the planes of polarization of the two waves are 
inclined to each other, the interference is diminished, 
and the fringes decrease in intensity; and that, finally, 
when the angle between these planes is a right angle , 
no fringes whatever are produced, and the waves no lon¬ 
ger interfere at all. These facts may be established by 
taking a plate of tourmaline which has been carefully 
worked to a uniform thickness, cutting it in two, and 
placing one-half in the path of each of the interfering 
waves. It will be thus found that the intensity of the 
fringes depends on the relative position of the axes of 
the tourmalines. When these axes are parallel, and con¬ 
sequently the two waves polarized in the same plane, 
the fringes are best defined; they decrease in intensity 
when the axes of the tourmalines are inclinedl to one 
another; and, finally, they vanish altogether -when these 
axes form a right angle. 


§ 161. The non-interference of waves, polarized in 




ELEMENTS OF OPTICS. 


353 


planes at right angles to one another, is a necessary result Experiments 
of the mechanical theory of transversal vibrations. In confirm the 
fact, it is a mathematical consequence of that theory, that theory of 
the intensity of the resultant light in that case is constant transversal 
and equal to the sum of the intensities of the two compo¬ 
nent waves, whatever be the phases of vibration in which 
they meet. 

But although the intensity of the light does not vary 
with the phase of the component vibrations, the character 
of the resulting vibration will. It appears from Equation 
(107), that two rectilinear and rectangular vibrations com¬ 
pose a single vibration, which will be also rectilinear 
when the phases of the component vibrations differ by an 
exact number of semi-wave lengths; that, in all other 
cases, the resulting vibration will be elliptic; and that 
the ellipse will become a circle , when the component 
vibrations have equal amplitudes, and the difference of ^ esults of th,s 

-*• x 1 theory and their 

their phases is an odd multiple of a quarter of a wave experimental 
length. These results have been completely confirmed by eonfirmation - 
experiment. 

In the above mentioned law we find the explanation of Apparent 
the fact, that no phenomena of interference or color are removed.^ 159 
produced, under ordinary circumstances, by the two 
waves which emerge from a crystalline plate,—for these 
waves are polarized in planes at right angles to one an¬ 
other ; and we see that, to produce the phenomena of 
color in perfection, the planes of polarization of the two 
waves must be brought to coincide by the analyzer. 


§ 162 . Fresnel and Arago discovered, further, that Law dednccd 

° , . . from experiment; 

two waves polarized in planes at right angles to each 
other, will not interfere, even when their planes of po¬ 
larization are made to coincide, unless they belong to a 
wave, the whole of which was originally polarized in one 
vlane / and that, in the interference of waves which had 
undergone double refraction, half a wave length must be 
supposed to be lost or gained , in passing from the ordi¬ 
nary to the extraordinary system,—-just as in the transi- 
23 



354- 


natural PHILOSOPHY. 


Another 
confirmation of 
the theory of 
transversal 
vibrations; 


Experiment 
detailed; 


Complementary 

colors. 


tion from the reflected to the transmitted system, in the 
colors formed by thin plates. 

The principle of the allowance of half a wave length 
is a beautiful and simple consequence of the theory of 
transversal vibrations. In fact, the vibration of the wave 
incident on the crystal is resolved into two within it, at 
right angles to one another,—one in the plane of prin¬ 
cipal section, and the other in a plane perpendicular to 
it. Each of these must be again resolved, in two fixed 
directions which are also perpendicular; and it will easily 
appear from the process of resolution, that, of the four 
components into which the original vibration is thus re¬ 
solved, the pair in one of the final directions must con¬ 
spire, while in the other, at right angles to it, they are 
opposed. Accordingly, if the vibrations of the one pair 
be regarded as coincident, those of the other must differ 
by half a ware length. Hence, when the plane of reflex¬ 
ion of the analyzer coincides successively with these two 
positions, the colors, which result from the interference 
of the portions in the plane of reflexion , those in the per¬ 
pendicular plane being not reflected, will be complemen¬ 
tary. 


Office of the 
polarizer; 


Explanation of 
appearances. 


§ 163. The former of the two laws explains the office 
of the polarizer in the phenomena. To account mechani¬ 
cally for the non-interference of the two waves, when 
the light incident upon the crystal is unpolarized, we 
may, § 133, regard a wave of common light as composed 
of two waves of equal intensity, polarized in planes at 
right angles to one another, and whose vibrations are 
therefore perpendicular. Each of these vibrations, when 
resolved into two within the crystal, and these two 
again resolved in the plane of reflexion of the ana¬ 
lyzer, will exhibit the phenomena of interference. But 
the amount of retardation will differ by half a wave length 
in the two cases; the tints produced will therefore be 
complementary, and the light resulting from their union 
will be white. 



ELEMENTS OF OPTICS. 


355 


§ 161. The preceding laws of interference being kept Reason of the 
in mind, the reason of all the phenomena is apparent. phenomcna; 
The wave is originally polarized in a single plane, by 
means of the polarizer; it is then resolved into two waves 
within the crystal, which are polarized in planes at right 
angles to each other; and these are finally reduced to 
the same plane by means of the analyzer. The two 
waves will, therefore, interfere, and the resulting tint will 
depend on the retardation of one of the waves behind 
the other, produced by the difference of the velocities Eesultant tinl 
with which they traverse the crystal. dependent upon; 

§165. It is plain, Equation (107), that the light issu- Emergent light, 
ing from the crystal is, in general, elliptically polarized, ^ e ° c e ^ is 
inasmuch as it is the resultant of two waves, in which polarized; 
the vibrations are at right angles, and differ in phase. 

Hence, when homogeneous light is used, and the emer¬ 
gent wave is analyzed with a double-refracting prism, 
the two waves into which it is divided vary in intensity 
as the prism is turned, neither, in general, ever vanish- 

,f ri . 7 . , . , ® J . . . When the 

mg, W hen, however, the thickness oi the crystal is such thickness gives 
that the difference of phase of the two waves is an exact the difference of 
semi-wave lengths , they will constitute a plane 
polarized wave at emergence,—the plane of polarization semi-wave 
either coinciding with the plane of primitive polarization, lengths ’ 
or making an equal angle with the principal section of the 
crystal on the other side, according as the difference of 
phase is an even or odd multiple of half a wave length. 
Accordingly, one of the waves into which the light is 
divided by the analyzing prism, will vanish in two posi¬ 
tions of its principal section; and it is manifest that the 
successive thicknesses of the crystalline plate, at which 
this takes place, form a series in arithmetical progres¬ 
sion. On the other hand, when the difference of phase 
is a quarter of a wave length , or an odd multiple of that Whgn tbe 
quantity,—and when, at the same time, the principal difference of 
section of the crystal is inclined at an angle of 45° to the phas ® 18 8 
plane of primitive polarization — the emergent light will be wave length, 


number of 



356 


NATURAL PHILOSOPHY. 


On'-uiar circularly polarized. This is one of the simplest means 
pelfectforone °f obtaining a circularly polarized wave; but it has the 
color only. disadvantage, that the required interval of phase can only 
be exact for waves of one particular length, and that, 
therefore, the circular polarization is perfect only for one 
particular color. 


Color may be 
produced with 
thick plates; 


Method 

explained. 


§ 166. We have seen that the phenomena of color are 
only produced when the crystalline plate is thin. In 
thick plates, where the difference of phase of the two 
waves contains a great many wave lengths, the tints of 
different orders come to be superposed (as in the case 
of Newton’s rings, where the thickness of the plate of 
air is considerable), and the resulting light is white. 
The phenomena of color may still, however, be produ¬ 
ced in thick plates, by superposing two of them in such 
a manner, that the wave v/hich has the greater velocity 
in the first shall have the less in the second. We have 
only to place the plates with their principal sections per¬ 
pendicular or parallel, according as the crystals to which 
they belong are of the same, or of opposite denomina¬ 
tions. Thus, if both the crystals be positive, or both 
negative, they are to be placed with their principal sec¬ 
tions perpendicular; and on the other hand, these sec¬ 
tions should be parallel, when one of the crystals is po¬ 
sitive and the other negative. The, reason of this is 
evident. 


Effects produced g 167. p e t us now cons ider the effects produced when 

when a polarized ° x 

wave traverses a a polarized wave traverses a uniaxal crystal, in various 
omaxaicrystal; di rec ti ons inclined to the axis at small angles; and let 
us suppose, for more simplicity, that the crystalline plate 
is cut in a direction perpendicular to the axis. 


Let AB 0 D be the 
plate, and E the place of 
the eye. The visible por¬ 
tion of the emergent beam 
will form a cone, A EB, 
whose vertex coincides 


Pig. 107. 










ELEMENTS OF OPTICS. 


357 


with the place of the eye, and axis E 0 , with the axis Ray coinciding 
of the crystal. The ray which traverses the crystal in wlth the 11X13 

^ ^ undergoes no 

the direction of the axis, P 0 E, will undergo no change change; 
whatever; and will consequently be reflected or not from 
the analyzing plate, according as the plane of reflexion 
there coincides with, or is perpendicular to, the plane 
of first reflexion. But the other rays composing the cone 
will be modified in their passage through the crystal, 
and the changes which they will undergo will depend on 
their inclination to the optical axis, and on the position 
of the principal section with respect to the plane of pri- other rays wm 
mitive polarization. 

Let the circle represent the sec¬ 
tion of the emergent cone of rays 
made by the surface A B of the 
crystal; and let MM' and N N\ 
be two lines drawn through its 
centre at right angles, being the 
intersection of the same surface 
by the plane of primitive polariza¬ 
tion, and by the perpendicular 
plane, respectively. Now, the vi¬ 
brations which emerge at any 
point of these lines will not be resolved into two within the Vibration8that 
crystal, nor will their places of polarization, that is, of win not be 
vibration, be altered ; because the principal section 0 f resolved; 


Fig. 108. 



Section or the 
emergent pencil 
by the face of 
the crystal; 


Fig. 109. 



Illustrations; 






vibrations that tp e crystal, for these vibrations, in the one case coincides 
resolved; 6 with the plane of primitive polarization, and in the other is 
perpendicular to it. These waves, therefore, will be reflect¬ 
ed, or not, from the analyzer, according as the plane of 
reflexion there coincides with, or is perpendicular to, the 
plane of first reflexion. In the latter case, a black cross 
will be displayed on the screen, and in the former a white 
one. 

But the case is different with the vibrations which 
win be resolved; emer g e an y other point, such as L. The principal 
section of the crystal for these vibrations, neither coincides 
with, nor is perpendicular to, the plane of primitive po- 


White or black 
cross. 


Vibratiens that 


larization; and consequent¬ 
ly the incident polarized 
wave will be resolved into 
two, within the crystal, 
whose planes of polariza¬ 
tion are respectively paral¬ 
lel and perpendicular to 
the principal section 0 L. 
The vibrations in these two 
waves are reduced to the 


Fig. 10T. 



same plane by means of the 


lieduced to the analyzer; they will, therefore, interfere, and the extent 
th^aifaiyze^and of that interference will depend upon their difference of * 

interfere. phase. 










ELEMENTS OF OPTICS. 


359 


Now, the difference of phase of the two wa ves varies Extent of 
with the interval of retardation. When this interval is lnterference 

dependent on 

an odd multiple of half a wave length, the two waves difference of 
will be in complete discordance; and, on the other hand, phase ' 
they will be in complete accordance, and will unite their 
strength, when the retardation is an even multiple of the 
same quantity. The successive dark and bright lines 
will, therefore, be arranged in circles. 

§ 168 . We have been speaking here of homogeneous Phenomena 
light. When white or compound light is used, the rings 
of different colors will be partially superposed, and the 
result will be a series of iris-colored rings separated by 
dark intervals. All the phenomena, in fact, with the ex¬ 
ception of the cross, are similar to those of Newton’s 
rings; and we now see that they are both cases of the 
same fertile principle,—the principle of interference. 

These rings are exhibited even in thick crystals, because 

the difference of the velocities of the two waves is very Analogous to 

small for rays slightly inclined to the optic axis. Newton snngs 


Fig. no. 




Illustrations; 


§ 169 . We will now consider briefly the case of biaxal 
crystals. Let a plate of such a crystal be £ut perpen- 






360 


NATURAL PHILOSOPHY. 


Effects of biaxal 
crystals. 


Lemniscat* and 
their 

fundamental 
property; 


Form of the 
dark brushes 
determined. 


dicularly to the line bisecting the optic axes, and let it 
be interposed, as before, between the polarizer and ana¬ 
lyzer. In this case, the bright and dark bands will no 
longer be disposed in circles, as in the former, but will 
form curves which are symmetrical with respect to the 
lines drawn from the eye in the direction of the twc 
axes. The points of the same band are those for which 
the interval of retardation of the two waves, is constant. 
The curve formed by each band is the Lemniscata of 
James Bernouilli,— the fundamental property of which 
is, that the product of the radii vectores, drawn from 
any point to two fixed poles, is a constant quantity. The 
exactness of this law has been verified, in the most com¬ 
plete manner, by the measurements of Sir John Her- 
schel. The constant varies from one curve to another,— 
being proportional to the interval of retardation, and in¬ 
creasing, therefore, as the numbers of the natural series 
for the successive dark bands ; for different plates of the 
same substance, the constant varies inversely as the thick¬ 
ness. 

The form of the dark brushes , which cross the entire 
system of rings, is determined by the law which governs 
the planes of polarization of the emergent waves. It 
may be shown that two such dark curves, in general, 
pass through each pole; and that they are rectangular 
hyperbolas t whose common centre is the middle point 1 of 
the line which connects the projections of the two axes. 


END OF OPTICS. 



APPENDIX 


No. I. 

Suppose a general wave front, sensibly plane, to have reached an open¬ 
ing AB , in a partition M N\ it is proposed 
to find the displacement which it will pro¬ 
duce in a molecule situated behind and 
anywhere, as at 0 , on the arc of a semi¬ 
circle M 0 iV, of which the plane is normal 
to the partition, and the centre at the mid¬ 
dle point of the opening. 

Take any molecule as Q ; draw 0 Q and 0 C\ make C 0 = r ; Q 0 = y\ 
C Q = z\ C A = b ; the angle 0 C Q = 6 ; and denote the whole dis¬ 
placement at 0 by i), then 

y = yV a — 2 r cos 6 z -f* z 2 , 
and by Maclaurin’s formula, 

. , sin 2 & , . 

y = r — cos & • z -f —- z 2 — &c..(a) 

2 r 



The displacement at 0, produced by the wave from Q, will, Eq. (19), be 
a • r, F t — y~| 

and from the molecules in the distance d z , 

adz . I" Vt — y~\ 

r a „i 2 f -rJ: 

and from those in the entire distance A B , 

_ r+ b a,'dz . Vt— 7 /“| . 

D =J. b -r -“L 2 '—x ~ J. (b) 

To facilitate the integration, suppose the greatest value of z to be very small 












362 


APPENDIX. 


as compared to r, and also the greatest displacements at 0 , by the par¬ 
tial waves from the molecules on A B , to be equal to one another, then wiil, 
Equation (a), 

y —T — cos Q z , 

and writing r for y in the coefficient of the circular function, Equation (b) 
becomes, 

D = sin ^ ( V t — r -f- cos 6 z) d z ; 

and performing the integration without regard to limits, 




2 irr cos 6 A 

and between the limits — b and -f b, 


a A 2tf 

cos— ( V t — r cos Q z ); 


_ a A [" 2 

I) =1 --- • cos — (Vt 

2*r cosy L A 


2 


r — cosdi) — cos^ ( Vt — r -f cos 6 6 )J 3 


or. 


a A . 2 b cos 1 


D =-- • sin 

<ir r cos 0 A 




This represents a displacement whose maximum is 


D. = 


a A 


at r cos d 


sin 


2<ir b cos 


(*) 


and which, therefore, determines the intensity of sound in air, or of light in 
ether. 

But this becomes zero for such values of 6, as make b • cos 6 ~ A, equal to 
either of the following numbers, viz: 



or which is the same thing, make b cos 6, equal to either of the quantities 


A 3A 5 A ? A , 

2’ it m in &c * 


So that, when the radius r is very great, in comparison with b , there will 
be upon the semicircular arc alternate places of sound or silence, light or 










APPENDIX. 


363 


darkness, symmetrically disposed upon either side of the point E, correspond¬ 
ing to which 6 is 90°. 

Sound decays rapidly as the distance it has travelled increases, and within 
the range of ordinary experience r cannot be very great. The relation 
assumed between r and b , to integrate Equation (b), can only be obtained, 
therefore, for audible sounds, by making b very small. And since X may 

be many feet, let us take the case in which the fraction ^ is so small as to 
justify the substitution of the arc 

2 * b cos 6 


in Equation (c), for its sine; in which case the intensity will be deter¬ 
mined by 

aX 2 <jc b cos 6 2a b 
' x r cos & X r ’ 

in other words, the sound passing through a small opening will be diffused 
with equal intensity in every direction behind the partition. 

Light follows the same law of decay as sound, but the value of X for the 
waves of ether being extremely small, the greatest not exceeding the 
0,0000266 of an inch, the limitations supposed with regard to the fraction 

-, in the case of sound, will not apply in that of light, and there must exist 
X 

the alternations of light and shade above referred to. 

When d approaches nearly to 90°, cos & will be exceedingly small, and 
the arc 2 if b cos & — X may again be substituted for its sine, in which case, 
Equation (c), 



which determines the intensity directly opposite the opening. The maxi¬ 
mum value for D, in Equation (c), will arise when 


2 . b . cos 6 


sin 


= ± 1 , 


which gives, Equation (< 


E/' = 


a X 

7 t r cos')* 


and as the intensity of light varies as the square of the greatest displace¬ 
ment, § 53, we have 






APPENDIX. 


m 


(d) 

Substituting the value of X for the longest wave of light, it is obvious that 
for any appreciable value for the cos 6, the intensity of light becomes insignifi¬ 
cant, and the only sensible illumination will be immediately opposite the 
opening. This explains the rectilinear propagation of light; and why it is, 
“ we may not see , and yet may hear around a corner 


“ 2X2 


v hence 


{D/'f = {D;y 


sr 2 r 2 cos 2 6 ’ 


X 2 


4 tf 2 . U 1 cos 2 6 


No. 11. 

Differentiating Equation (11), we have 



differentiating Equations (3) and (3)', we obtain from them 

d _ cos 9 cos 4 j d 4 ,' 
d 9 cos 9 ' cos 4 ' d 9 '* 

and from Equation ( 10 ), 

d<?'~ ’ 

and this, combined with Equations (a) and (b), gives 

cos 9 cos 4 / 
cos 9 ' cos 4 *' 1 

which will be satisfied by making 


• « 


• • (b) 


9 = 4'; 9 ' =4 / - 

That is, the deviation becomes a minimum when the angles of incidence 
and of emergence are equal. 












APPENDIX. 


365 


No. III. 

Differentiating Equation (104), we find 


d_8 
d 9 


= q=[2-2(» + l)^] = 0;. 


and from Equation (105), 


d<p' 


cos 


dcp m. cos 9 '* 
which substituted above, gives, 


=F [”2 — 2 (n + l)-52I2_j = 0 ; 
L v ’m cos 9 'J 


whence 


cos o 


n + 1 m cos 9 ' 


which is the first equation of § 126. 

Differentiating Equation (a) again, we find 


= * p — j - — • • ( m2 cos2 <?' - cos 2 9) ]r 

d a? 2 L m 2 cos 3 p .1 


(») 


and since 9 > 9 ', cos 9 < cos 9 ', therefore the last factor must be positive ; 
whence 8 is a maximum in the primary, and a minimum in the secondary 
bow. 






















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